Is Weird Al Yankovic a superstar? Well, he is probably the only 55-year-old nerd in poodle curls and a Hawaiian shirt who can hold aloft an accordion, rock-god style, and be greeted by screams from women and hollers of "We love you, Al!" from grown men. And this is in Las Vegas on a Friday night, with Britney Spears playing the arena next door.

Certainly, Yankovic is a phenomenon. He has sold over 12 million albums and won four Grammys for his pop song parodies such as "Eat It" – Michael Jackson's "Beat It" with lyrics about finishing one's dinner – and "Amish Paradise" – a spoof on Coolio's "Gangsta's Paradise". ("And I've been milkin' and ploughin' so long that/ Even Ezekiel thinks that my mind is gone.") In American culture he is a perennial, peculiar touchstone, popping up everywhere from Scooby-Doo to Friends, The Simpsons to Veep. Chamillionaire, whose "Ridin'" was spoofed by Yankovic in his platinum-selling "White and Nerdy" ("They see me mowin'/ My front lawn…"), thanked the parody for helping him to win a Grammy. "Because it made the record so big it was undeniable. It was so big overseas people were telling me they had heard my version of Weird Al's song."

Last year, Yankovic's 14th studio album, Mandatory Fun, became the first by a comedian to go to No 1 on the Billboard charts since Allan Sherman's My Son, the Nut in 1963. One track, "Word Crimes", an attack on sloppy grammar set to "Blurred Lines", also made the singles chart, making him only the third artist to have a Top 40 hit in each of the last four decades. The other two are Michael Jackson and Madonna.

So yes, Weird Al is, weirdly, a superstar. "That's the irony of my career," he says the morning after the premiere of his new world tour in Las Vegas. He is wearing jeans, Vans trainers and a canary-yellow CIA T-shirt, a look that clashes with the plush purple carpets, black walls and chandeliers of the high-roller suite at Planet Hollywood where he is currently staying. "I've always considered myself an outsider, poking fun at the people in the inner circle. And now, after all this time, I've acquired enough fame that I feel like I'm in the inner circle with the people I'm making fun of. So it's kinda strange."

The Mandatory world tour lands in the UK next week and will see Yankovic and band perform old favourites like "Eat It" and "Smells like Nirvana" ("What is this song/ All about/ Can't figure any/ Lyrics out") alongside more recent hits like Iggy Azalea's "Fancy", retooled as "Handy", a hymn to DIY, "Word Crimes" and Miley Cyrus done polka-style on an accordion. There are daft costumes throughout, from an upside down ice-cream cone for Lady Gaga, to a fat suit for his take on Michael Jackson's "Bad", "Fat". "I like to say I have three more costume changes than Lady Gaga", he says, with a high-pitched giggle.

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On the night I saw the show, the audience ranged from children to pensioners, and they all sang along to every word. "I think I appeal to the 12-year old boy in people", says Yankovic, who has an adolescent air himself and seems barely to have aged in three decades. It is good, clean, daft fun – no smut, no swearing. "It's an extension of my personality. I don't even use profanity in everyday life and I wouldn't put it in a song. It's the way I was raised. When I first started recording things, I didn't want to do anything that my parents would find offensive."

Albert Yankovic grew up in Lynwood, a suburb of Los Angeles. When he was seven years old his parents enrolled him in accordion lessons. He is still not sure why but thinks they may have been inspired by the polka player, Frankie Yankovic (no relation) who was somewhat famous at the time.

They were a close, "perfectly happy", family but in 2004, tragedy struck when Yankovic's parents were found dead at home, from suspected carbon monoxide poisoning. "It was the most horrific moment of my life," he says. He was in the middle of a tour at the time and refused to cancel a single gig. "I'm a show-must-go-on type of person. I was going through a tragedy but I didn't want to stop the train because there were so many people counting on me. Even though it was extremely hard for me to do, I got up on stage every night for two hours and did a comedy show. During those two hours I was in complete denial. And in a way it was cathartic, it kind of got me through it. A lot of people over the years tell me that my music has gotten them through some very difficult times in their life and I thought, well maybe my music can do the same thing for me."

It's a tale that indicates a steeliness behind the silliness. But then Yankovic has been in showbiz since he was 16 years old when he sent a homemade tape into Dr Demento's radio show and got on air. He went on to study architecture at university but spent most of his time playing accordion in coffee houses and DJing on the college radio station, under the name Weird Al. "Probably when people were calling me weird at first, it wasn't in the kindest way," he says with his trademark chirpiness. "But I decided to take it on and be proud of my weirdness."

In his senior year, The Knack's "My Sharona" was riding high in the charts so Yankovic recorded his own version, "My Bologna", in the toilets at the radio station. Dr Demento played it, The Knack's lead singer heard it and told his label to release it. And so the parodies proper began.

How does he come up with them? "It's pretty rare that I'll spontaneously have a brilliant idea for something. It's happened maybe three times," he giggles. "More often than not, it's a very analytical process. I'm ostensibly doing all this silly stuff but I'm very serious about the craft of it."

As soon as he has a song in his sights, he asks permission from the artist. "I don't want to spend weeks of my life working on something only to find out that that artist has no sense of humour." Not everyone gets Weird Al. Prince stopped taking his calls back in the Nineties. Eminem said no to a video in case it harmed his rep. Coolio was furious about "Amish Paradise" (but finally admitted it was "funny as shit"). And then there was the time Paul McCartney said no to his version of "Live and Let Die", or "Chicken Pot Pie."

"He said 'If you could make it about anything other than chicken, great, but I'm a strict vegetarian and I don't want to condone the eating of animal flesh…' I couldn't figure out any other way to make it work, so I just let it go."

Most artists consider a Weird Al parody a badge of honour. "Lady Gaga called it a rite of passage," he says proudly. Not one person turned him down for his last album and he has some friends in high places. Mark Knopfler plays on "Money for Nothing/ Beverly Hillbillies*". It was Madonna who suggested "Like a Surgeon" as a fun parody of "Like a Virgin". And Michael Jackson allowed him to use his own film set to shoot "Fat". He must be quite the charmer. "Here's the thing. Pretty much all my humour is not mean-spirited," says Yankovic. "My parodies are all done in good fun. I don't try to step on anybody's toes, I'm not denigrating any artist, I'm not out for blood, I'm just out to have a good time. I like to say that it's more a poke in the ribs than a kick in the butt."

Perhaps the most remarkable thing about Yankovic is that he has endured. These days, anyone can make a music parody and put it online within hours of the song being released – and many do. So Yankovic decided to play them at their own game. When his record label told him there was no marketing budget for Mandatory Fun, he decided to do a Beyoncé – and release eight songs and videos online in eight days. His genius stroke was to partner with various websites, who got online exclusives in return for their support. His parody of Pharrell's "Happy" – "Tacky" – went to Nerdist, Lorde spoof "Foil" to CollegeHumor, "Word Crimes" to Vevo and so on. "I started thinking, I don't need a hit single, what I need is to dominate the conversation for a week. I need to be omnipresent on the internet for the entire week of release."

He was and it worked but he remains pessimistic about the record business. Mandatory Fun is the biggest album of his career. "It's huge but it has not even gone gold. It's tough. I don't want to be complaining. I've got a nice career, a nice life and I'm doing just fine, thank you. But I'm not making the kind of money that some people might think I'm making." His next step is likely to be a Broadway musical. "This is probably going to be my last conventional album, which sounds kind of insane after having my first No 1. The material that I do has to be timely and topical and that's difficult if you have to wait around to get 12 songs released all at once."

He lives in LA with his wife, Suzanne, a former executive at 20th Century Fox and their daughter, Nina, who is 12 and learning the accordion. "I love that she doesn't follow the crowd, she's her own thinker. You never know what's going to rub off. A couple of years ago all of her friends in school were obsessed with One Direction. She said, 'Yeah they're OK but I like Glen Campbell'. Wow! That's random." It is, he says, "a pretty quiet, non-weird family life", though they do have a second home in Maui that they escape to when possible.

Does it feel odd to have hit the big time in his fifties? "I would have loved it in my twenties, thirties or forties, of course, but having it at this point in my career is extremely gratifying. A lot of the people who have been in the business for 30-plus years are nostalgia acts. They're not peaking. It makes it that much sweeter."