If you need to tell people to relax and have fun ("Mandatory fun day"), it's an indication that you have a more fundamental problem. It's probably better to address that instead of making playful signs.
I think you're missing the point. This is not about "mandatory fun", it's about recognizing the runaway restrictiveness of public parks and historical sites that has resulted in people expecting that common activities will be prohibited. It's poking fun at the uptight reputation that such places have, and attempting to reverse the trend by loosening those irrational restrictions.
Exactly. If I went to a place such as the one with the "Keep on the grass" sign, and didn't see any signs, I would actually pretty much assume that anyway. Because it's just so common that I expect it to be the default stance at such locations.