But by the late 1980s, more shows wanted to distance themselves "from the corny, syrupy stuff" -- and in stepped shows such as "Married With Children" and "The Simpsons."

It's only natural that comedies would take on dads, Thompson says.

"Comedy is about inversion -- taking people who are in authority and control and making them the butt of jokes." So in a society "that has been so dominated by men... comedy is naturally going to play against that."

Dads on TV today are presented in so many different ways that it's impossible to say things are getting better or worse, he says.

From the stay-at-home dad in "Up All Night" to the lovable dads in "Modern Family," relatively positive portrayals abound. Thompson says the growing push for "recognizable" dads who reflect today's realities are probably to credit for these characters.

Meanwhile, dramas such as "Game of Thrones" and "Mad Men" show such bad fathers that "you'd beg social services" to put their children in a foster home with Al Bundy, he jokes.

Plus, thanks to modern media, people now can always create their own videos about dads and, sometimes, watch them go viral.

To some, the idea that anyone would be upset over this stereotype is ridiculous.

Huggies' Facebook page has hundreds of posts from women and men saying they found the original ads funny. "The problem these days is everyone has lost their sense of humor," Christine Lee wrote.

And every group of people -- including moms -- gets caricatured and stereotyped on TV.

Do some dads just need to lighten up?

No, says Routly. There's a growing rejection of stereotypes in general -- and that should apply to dads as well, he argues.

He's also concerned that boys and men "see the bumbling dad ... and think that's what's expected of them," the stay-at-home father of two told me by phone while baking chips for his kids out of kale from his garden. "They're not expected to be good, so they rise to the low bar that's set for them."

And it can lead girls and women to have low expectations for how their husbands will handle fatherhood, he says.

Part of the problem, Routly says, is that a lot of people believe negative stereotypes really apply to far more dads than they do.

I ran into that myself last year. A column on CNN.com by a stay-at-home father complained that "most" dads "just wanna chill in front of SportsCenter with a bowl of chips" after work and complain to their stay-at-home wives about how hard they work. On his blog, the writer railed against the "millions of dads who view their days at home as recovery from work."

I responded with a long list of studies showing how well dads are actually doing, including this from Pew Research: "Almost all fathers who live with their children take an active role in their day-to-day lives through activities such as sharing meals, helping with homework and playing."

While by far most of the responses I received were positive, some were from angry people insisting the negative depictions must accurately describe the average dad.

Routly says he received hate mail after launching his petition. "There are people who are so invested in maintaining these kinds of roles and stereotypes that they felt like they needed to attack me."

But he also heard from men thanking him and sharing stories of custody battles and judges who assumed they weren't as good at parenting simply because they're men.