On Father’s Day, my Instagram glowed with vintage portraits of my acquaintances’ dads, often in shorts and Freudian tan, daughters, now in their 30s, reframing them in Tinder proportions as if men, actual men. Even after the Father’s Day barbecues had cooled, the idea of dads as aspirational sex objects lingered. At last week’s Balenciaga show, its designer, Demna Gvasalia, was inspired by “young dads in the park with their kids at the weekend”, and cast real-life fathers as models in baggy blazers and denim. They stormed down the Paris catwalk dragging their bemused toddlers as if exiting Tesco after an incident in baked goods.

It shouldn’t be surprising that dads have become sexy. Somebody has definitely had sex with them at least once

I mean, it shouldn’t be surprising that dads have become sexy. Somebody has definitely had sex with them at least once – the proof is in the yogurt on their collars, their single carrying muscle. But still, what a twist, no? That those weekend park dads could be the single most fashionable icons of our time?

Those weekend park dads, bellies pink and just slightly visible between their belt and the bottom of their Corbyn T-shirts. Eyes as flat as their Samsung phone, the screensaver their five-year-old grinning, three months ago, 6am. There she is on the swing now, feeling as if she’s flying, and her dad over here battling with desire. Not for women, God no, but for a glance at his phone – all the news he’s missing right now, sitting on this bench in this rubberised playground in this green city park. He’s given himself a target – 10 minutes without looking. But he’ll be honest, it’s really hard, man. And right now, if it was possible to wish away these precious minutes of his kids’ childhoods, then he would leap to the second when he could do that swift check, email, messages, news, social updates, email again, news quickly. There was a moment yesterday when he would have gladly extended his mortgage, just for quarter of an hour alone on the sofa to blindly stalk his enemies on Instagram. Funny though, when he found that time, when he stuck on Moana and lay back on the big cushion, he ended up just scrolling through pictures of the children. He loves them even more behind glass.

Those weekend park dads, a night out at the pub planned three months in advance, combing his hair in a mirror steamed up by the kids in the bath. He sprays himself vigorously, and booms a goodbye. And it’s not until the children are leaping on his wife, asking for jammy toast, that she phones him, blurrily. Answers on the first ring with the powdered jollity of a man who’s forgotten how to sleep. Wicked night, he screams, until about one, when he lost his left shoe. Anyway, long story short, he’s in Brighton. Student halls, nice bunch of guys, sort of, they’ve been watching Natural World for four hours, he cried at the baby kangaroos, styled it out, don’t worry. What’s the problem, he asks. You knew I was going out. Leaving now, anyway. Back in six hours.

Those weekend park dads who were once in a band that exists now only as a haircut. Those ones who go to rehab at Clowntown. Those weekend park dads who discuss the spec of the swing set. Those who always win against their eight-year-old. Those weekend park dads who like to earnestly teach you how to cook pasta “with a bite”. Those weekend park dads who use the playground as their private muscle gym. Those with a soft play tan. Those weekend park dads with big plans of how to overthrow the patriarchy before their daughter turns 12.

Those ones on their third marriage and sixth child, determined to know him, this time. Those ones who watch the sports on their iPad while their child dangles from a tree. Those ones still floating on a compliment from their 20s. Those weekend park dads who have their own mini-scooters, I think I love them most of all.

This idea of sexiness works because, rather than the swagger and attitude of a six-packed teenager, the idea of a weekend park dad is flawed and tired. It has cracks in it, into which one can stuff all the horror and fantasy of reality. This season, every day is Father’s Day.

Email Eva at e.wiseman@observer.co.uk or follow her on Twitter @EvaWiseman

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