Something strange happened recently to Trey Parker and Matt Stone. Of course, the pair – two of America's wisest and most puerile satirists – are fairly used to strange things happening. There was that time 20 years ago when they made a little cartoon video about a fight between Santa Claus and Jesus. It promptly went viral and landed them the contract for their long-running cartoon series South Park, which now nets them TV deals in the high eight-figure neighbourhood.

Now a "weird idea" they had has conquered Broadway. Just under a decade ago, they decided to make a musical about Mormonism.

"Hollywood people would be like, 'Why the hell would you do that?'" Stone recalls leaning back in his seat languidly, though his words come out at a jittery speed. "And we'd be like, 'Because it's fucking funny!' We thought, 'Look, we won't make any money but if the show runs for a year, that would be, wow.'"

At times, it is hard to make out what Stone is saying, because next door to the midtown New York studio where we meet, Parker is training the third cast for The Book Of Mormon. The hit show is on tour in Chicago, alongside the current production in New York. Cheerful songs reverberate through the wall with lyrics such as "Fuck you, God, in the ass, mouth and cunt!" (Cole Porter it is not). Parker and Stone's weird idea, since it opened in 2010, has won nine Tonys, will soon come to London and has been credited with rescuing the modern musical. "This has been," Stone says, "like, oh my God."

Tom Cruise in South Park. Photograph: Wenn

Even harder to fathom: Sean Penn went to see The Book Of Mormon. Penn, Stone and Parker have been engaged in a war of expletives ever since the pair satirised the actor's doe-eyed idealism in their puppet musical, Team America: World Police (2004). In that film, which made fun of warmongering conservatives and self-righteous celebrity liberals alike, Penn in puppet form was held up for special ridicule.

Penn's characteristically measured response was to write Parker and Stone a letter signed, "All the best, and a sincere fuck you." But now, according to the US press, Penn has not only seen the musical written by the pair who once described him as "a giant douchebag with legs running around Haiti", but gone to see it twice. And loved it.

Parker is the more sweetly childlike of the pair, heavier set, with a boyish smile. He slips in and out of the room between rehearsals. Stone is the sensible one, who had planned on a career involving mathematics. For lunch, Stone has a healthy-looking soup while Parker opts for a McDonald's, getting sauce all over his face. Parker, characteristically, refuses to let the Penn row slide: "You know, if I'd seen him there… Sean Penn is one of the few people who I'd go up to and say, 'Fuck you,'" he says, grinning.

The Book Of Mormon, a larky buddy story about two American Mormons who go to Uganda in an attempt to convert the natives, is utterly irresistible. I've seen it twice (and bought tickets for the London production), and both times the whole audience, from celebrities to elderly couples, sang along to the profanest of lyrics, smuggled in good ol' fashioned Broadway tunes (the show is packed with winks to the classics, from The Sound Of Music to The Music Man).

Even those who have been at the sharpest end of the pair's satire can't fight their appeal. Barbra Streisand is reported to have attended the show, despite being portrayed in South Park as a demented robot who emits toxic gas from her nipples. (The possible exception to this rule is Tom Cruise: an infamous South Park episode mocking Scientology saw Cruise hiding in the wardrobe of a little boy, and the line, "Tom Cruise won't come out of the closet" was repeated more than 20 times. It is unlikely Cruise has been to see The Book Of Mormon.)

Parker and Stone are on no one's side. In the 15 years South Park has been on cable TV, it has been accused of being too conservative, too liberal, too amoral, too moral, too sophisticated and too crude. It is also, repeatedly, cited as one of the greatest television shows of all time and has won a prestigious Peabody award. It has been criticised by the Gay & Lesbian Alliance Against Defamation (Glaad) for its use of the word "fag" ("They missed the point," Stone grumbles), and nominated for a Glaad award for outstanding TV for its episode titled Big Gay Al's Big Gay Boat Ride.

"Everyone likes you when you're beating up on someone else, then when you turn on them they're like, 'What the fuck?'" laughs Parker.

Stone and Parker grew up in small Colorado towns similar to the one depicted in South Park. They met at the state university and were, by all accounts, the funniest guys on campus. They still have a tendency to finish each other's jokes, delightedly foreseeing the other's punchlines.

"When you grow up in Colorado, you don't think, 'I'll write a Broadway musical!' Or, 'I'll go to Hollywood and make a TV show!' That just doesn't happen," Stone says. "But in the 1990s, that was the heyday of independent films, so we thought, 'OK, we can make independent films and still live in Colorado.'"

After going to film school and jobbing around in their early 20s, they made the Santa and Jesus video that got them a contract at Comedy Central to make South Park, a cartoon series about four foul-mouthed 10-year-olds that contains more wisdom in one episode than some works of literature, and a lot more jokes about anal sex and the ridiculousness of Kanye West. (They mocked the singer's oversensitivity; West, not a man known for his self-awareness, responded by penning lyrics about how much South Park hurt his feelings. "You just want to say, 'Dude, look at yourself. Look at yourself,'" Stone says with an eye-roll.)

Stories abounded in the past two decades of Parker and Stone's frattish, brattish behaviour, from farting on colleagues' food to infamously attending the 2000 Oscars dressed as Gwyneth Paltrow and Jennifer Lopez while tripping on acid (their song, Blame Canada, from South Park: Bigger, Longer And Uncut, was nominated; it lost and they walked out). But today, Parker, 43, and Stone, 41, are a delight: friendly, warm, with none of the bullying or confrontational attitude that colleagues of old had reported.

That said, I wouldn't be surprised if a British journalist turns up in a future South Park episode, asking annoying questions and ending up getting mangled in some terrible accident. As one of their colleagues said in a 2011 New York magazine profile of the pair, "They're not snotty until you leave the room."

They have been described as both nihilists and libertarians, suggesting that to make fun of everything is to believe in nothing (or everything). Yet, beneath the simplistic animation and jokes about Hillary Clinton hiding a nuclear device in her vagina (Parker still sniggers at the memory of that episode), South Park is a deeply moral show. Parker and Stone almost never vote, "because we're not allied to one particular side, so it feels painful", Stone says, but it's safe to assume that they veer more towards the left in their attitudes (in one South Park episode, Satan muses how he can win over the people on a particular issue. "I'll do what we always do," one of his assistants replies. "Use the Republicans.") Yet of all the demographics they've teased in their time, the one that has given them the most grief is liberals.

"The big lie of our whole career is that rightwing fundamentalists are always trying to shut us down," Stone says. "It has literally never happened. The Mormons haven't, the Christians haven't – OK, the Scientologists did, but they don't count. But when we make fun of liberal people, they're like, 'What?!' I think religious conservatives are more used to taking a beating." And as if to antagonise their long-suffering liberal fans even more, Parker announces that the two causes he and Stone believe in are "gay marriage and guns. We're for both of those." [This interview was conducted before the recent shootings in Sandy Hook].

"We're from Colorado, and look at the way Colorado's gone politically in the last few elections," adds Stone, "it's now gay-friendly, weed-friendly, gun-friendly. There's an element of Colorado that I think is in us." Parker nods stoutly.

Whether they're taking on American healthcare or religious tolerance, Parker and Stone overturn expectations hilariously but thoughtfully, and often with a sweeter aftertaste than you expect.

If there is an overall message to their work, it is, be tolerant and be temperate, because only blinkered idiots are entirely on one side or the other. (Another general message, one thoroughly explored in Team America, is that "if you're famous, you suck, just for being famous", Parker says. "People in England totally get that; Americans don't.")

Chef, (behind), in South Park. Photograph: Reuters

They've never written something they regret exactly, but one episode of South Park provoked a reaction they regretted: when they mocked Scientology's avariciousness and sci-fi theology, Isaac Hayes, a devout Scientologist and the voice of the South Park character Chef, quit the show.

"I said to Isaac when he came in to see me, 'Look, my mother's Jewish and we've gone there, so come on, man.' It's all got to be OK or none of it is OK," Stone says with a shrug.

But Parker still sounds forlorn: "We just felt sad for him, because he was caught in the middle of two things he loved, and he was a friend. He really loved South Park and he really loved Scientology, and I think he felt he had to quit."

After Hayes left, Stone and Parker wrote an episode claiming Chef had been brainwashed, absolving the character of blame and urging the people of South Park instead to "be mad at that fruity little club for scrambling his brains". Two years later, Hayes died and Stone and Parker dedicated a show to him.

While their message is moderate, their comedy is decidedly not. South Park is the show, after all, that once featured Paris Hilton opening a chain of stores called Stupid Spoiled Whore, and The Book Of Mormon includes a scene in hell in which Satan gives Hitler a blow job while giant Starbucks cups dance in the foreground. They are utterly fearless in their attacks and are now pretty much allowed to say and do whatever they want.

"Comedy for me has to be either completely absurdist, or it has to be meaty and dark," Stone says. "I just can't do with romantic comedy. Really, you're going to do another joke about going on a date? I'm like, 'How do you go to work and do that?' You're not touching anything real, anything dangerous."

The Book Of Mormon conquered Broadway. Photograph: AP

The Book Of Mormon, which they wrote with Robert Lopez (Avenue Q), exemplifies the pair's skill at undercutting audience assumptions. Far from simply mocking Mormonism, it celebrates the human need for myths to make sense of the world, even if quite a few Mormon myths get a proper kicking: "I belieeeeve," one Mormon character croons in the show, "that in 1978 God changed his mind about black people!"

Parker and Stone have always felt a bemused fondness for those in the faith, if not necessarily for the faith itself (they call the show "an atheist love letter to religion"). The lead character in their first film, Orgazmo, was a sweet Mormon who acted in porn films to fund his upcoming wedding. (It was not, to be honest, their finest hour.)

"Mormons have this naive hopefulness which I find commendable," Stone says. "Then they start talking to you and you're like, 'What the fuck?' But I'm an atheist, and all religions sound pretty goofy to me. I think, really, at 2am, we all believe in some goofy shit."

They're glad that America's most famous Mormon, Mitt Romney, didn't win the election, but only because they think he wouldn't have made a good president, not because of his religion. When I ask if they saw the widely circulated five-minute clip of Romney appearing to flip out about his religion on a radio show, Stone admonishes me for having watched only the shortened version as opposed to the full 20-minute one. "In the original clip, it's the other guy who's really pushing him. Someone cut it to make Romney look bad, which I thought was really shitty."

Given their impatience with simplistic absolutes, it is unsurprising that the pair are no fans of certain prominent fellow atheists. "I'm counter-influenced by the neo-atheists, Richard Dawkins and those guys," Stone says. "I'm not convinced that truth is the most important thing in the world. Humans tell stories – that's what happens. I don't get [Dawkins'] trip." This is something of an understatement: when Dawkins made an appearance in South Park he was portrayed as an intolerant snob who was so unobservant he didn't notice the person he was sleeping with was a man. (Dawkins, to his credit, merely complained they got his English accent wrong.)

Perhaps the strangest thing that's happened to Stone and Parker is that, after years of being much beloved cult favourites, they are at risk of being taken in by the mainstream, all thanks to a musical about Mormons. "When people say, 'Will middle-aged people from middle America like this?' we're like, 'Middle-aged people from middle America made this,'" Parker says.

Both have talked in the past about their fear of losing their edge with age, and they are in settled relationships. Stone has two children under three and Parker has one stepchild.

Yet South Park, which has been commissioned for another four series, feels angrier than ever and their side projects – from Team America to The Book Of Mormon – get more sophisticated.

"I thought I'd become a fucking softie when I had a kid," Stone says, "but that doesn't seem to have happened."

Despite being wealthy enough to delegate, or to stop working altogether, they are as hands-on with South Park as when they started – writing, voicing and directing it. They will be personally involved with The Book Of Mormon for as long as it runs, moving their families from LA to England for the London opening at the end of next month.

Partly this comes from a sense of pride in their work, which has not dulled with wealth, and their fear of "fucking it all up" by letting any of their shows go slack. But mainly it comes from loving a job that "pretty much lets us do whatever we want", Stone says, sounding geeky with wonderment that they still do what they do.

"It's weird, because in many ways what we do now is the same as what we did when we were 21. We're just trying to make each other laugh."

• The Book Of Mormon is due to open in London on 25 February 2013 [this was corrected, from 25 February 2012, on 7 January 2013].

• This article was amended on 8 January 2013 to make clear that the Book of Mormon has already opened in Chicago.