Actually, cyberspook, the press does not take pictures of children "all the time" or, for that matter, even routinely - not without parental consent and releases, or not without an extremely good newsworthy reason to do so. In addition, photojournalists' cameras are generally pretty hard to ignore, so surreptitious photos are not taken (with a few, very rare, investigative-type exceptions).
I will give you some reasons why I think your plan is dangerous legally.
First of all, taking photos of people in public places is generally considered legal, and this includes taking pictures of children in public places. The best way to protect oneself from being sued, however, is to obtain permission prior to taking a photo, even when permission is not legally required. For consent to be effective, it must be given by someone capable of giving such consent. When photographing a minor, this means getting permission from the parent.
A person with a camera taking photos of children in a public place is often regarded with suspicion. All it takes is one concerned person to report to the police and the camera could be confiscated. The taking of photos surreptitiously using a cell phone camera is regarded with even more suspicion and, if discovered and reported, the cell phone would most certainly be confiscated.
If any photo taken captures, even in all innocence, a child's "private area", or if any photo taken could be considered "sexually arousing" in any way, shape or form, major legal woes await.
The 2004 Video Voyeurism Act was amended to include public areas, and many states now have electronic voyeurism privacy laws, designed in part to address the increasing problems with "down-blousing" and "up-skirting" occurring with cell phone camera use in public areas. A Predator Alert Act, introduced in Congress in September of last year, has a good chance of being enacted this year - this will require cell phone cameras to sound a tone when a photo is being taken. California has a similar law that will take effect this year, I believe. This is how seriously cell phone camera use is being questioned, especially as it relates to their use in photographing random children.
Now, even if you were to surreptitiously take a photo with your cell phone of a kid doing something "bad" and you were not discovered taking the photo, posting (publishing) this photo on a "wall of shame" opens you up to potential defamation actions. If the photo is of a recognizable subject and displayed in an unfavorable way, or even in an embarrassing way, or if the photo implies something unfavorable, this can be defamatory or at least bring about a defamation action. I hope you realize that many parents WILL sue over something like this, even when these same parents are not concerned about the reason WHY their child's photo was posted in the first place.
In addition, there is a chance that, in addition to defamation, an invasion of privacy false light action could arise from the photo-taking, photo-publishing, should you mistakenly photograph an innocent act and/or innocent child and post it on your "wall of shame".
So, even if your photo taking in itself is legal, and even if your photo-board shows kids whom you can prove are trouble-makers and worthy of a "wall of shame" position, that not only does not prevent a lawsuit, it can very well result in one. And you would find yourself having to defend your actions in court.
I suppose you could inform all park-goers in some way that the entire park is under survellience and that photos may be taken of anyone in the park, but I am not sure the park service will allow for this. And this would not prevent a passer-by from calling the police and saying you were taking photos of children, and it would not eliminate the risk that one photo might be considered improper, or that one photo might be considered defamatory.
Your plan is certainly not one I would risk, cyberspook. If you are having problems with the kids in the park, I suggest you call the police and let them handle the matter. Or contact the park service and have REAL surveillance cameras installed.