"It's Time To Stop Treating Dads Like Idiots" - from DaddyFiles.com
The article linked talks about the author's response to the onesie pictured, and how this idea of mocking fathers is both prevalent and bad. I agree.
I think I'm lucky in that I haven't had this kind of thing targeted directly at me, but it's very true that pop culture sets abominable examples for fatherhood. It's something I was aware of, even from a relatively young age (or so I remember). On the one hand you've got the completely and disgustingly incompetent Homer, the spineless and emasculated Ross, the well-meaning but useless father figure Disney loves to show (Aladdin being a prime example), the borderline abusive who is emotionally unavailable (Red Forman), the absentee and/or deadbeat and/or left his family to marry a hotter and younger woman and no longer has any interest in his child (pick just about any 80's movie), and of course the Actually Evil (Darth Vader). Even when fathers ARE competent (which is rare), they usually die or are otherwise made unavailable (Mufasa, Uncle Ben).
On the surface it might seem like that is changing, with the image of the still-cool dad with the baby carrier on his chest become more prevalent, but I still haven't seen one of those that doesn't come with a slight edge of mockery or tacit encouragement that being a "cool dad" means you still behave like a frat boy when the Overbearing Wife isn't looking. I saw an ad for a show about fatherhood that featured four dads sitting in a playground ignoring their children while drinking and smoking pot. Absolutely maddening.
That relationship with the wife can also be a pretty toxic message. Assuming she's even AROUND (I'm looking at you, Disney, AGAIN). Frequently wives and mothers are seen as the spoilers of fun and cleaners of messes who do all the "real" parenting while Dad learns Important Lessons and doesn't do much beyond delivering the occasional heart-to-heart at a critical juncture. Frequently this Dad Ex Machina kind of ending or plot point seems to make unimportant all the work the so-overworked-she's-bitchy mother and reinforce the message that fatherhood is not really something you do all the time.
Another example of this phenomenon of casually mocking fathers, to return to the original point of the linked article, is the meme to the right, of "fathers in the baby food aisle" that shows several men on their phones clearly getting their wives to guide them through purchasing the correct baby food. This picture really irritated me, and it's because there's a certain element of truth in it. The reality is that I do not spend all day, every day with my son and I am not the primary decision-maker on many child-related products, so I sometimes have to check with the wife before I get stuff. This notion that I am somehow failing or making myself worthy of mockery is frankly shitty. I can't deny that I sometimes feel self-conscious when buying baby products on my own, especially if I have to take a moment to find or decide between things and it's for exactly this reason. I have more than once gambled on getting the wrong thing before being seen to call for help because I KNOW that that's how the parents around would view me. So the danger there is that why try and be better, why risk failing when the reward is not education or support but negative reinforcement?
This is not, of course, to suggest that there is not an equivalent cultural phenomenon targeted at women. There is, it just applies its pressure and sets its examples on the other end of the spectrum. The difference, I find, is that there is also a cultural protectionism of mothers. You can criticize imperfection in mothering, but you can't mock just for trying. Fathers can be both criticized and mocked, and are. You're criticized if you don't get involved and be an active parent, and you're mocked if you do.
All of that said, though, the world is not entirely a carnival of feces and shame. As I said above, I'm fortunate in not being subjected to this on a personal level, and as my peer group is increasingly populated by parents I see how many fathers there who appear to be doing a great job despite the dearth of positive examples in the media. I suspect internet culture has a great deal to do with this, as resources like the linked-to blog proliferate.
I would also be remiss if I didn't point out that I had a pretty good role model for how To Be A Dad. That's always a bonus.