As a meeting place, maybe access could be a problem. Depending on the kind of situations you're thinking of, things like flooding might be more likely on the way to a wildnerness area than around town, perhaps making it harder to meet up safely or in a timely manner.

Oh, the residents that live around the park may be very friendly and helpful, but then again, they may be very suspicious or even hostile to "outsiders". Even the authorities may be this way. I'm sure that I'm not the only one who read news stories of some people fleeing the flooding in New Orleans, only to be turned back by police roadblocks at the border of neighboring parishes. It's possible that roads to state parks will be blocked off by the park rangers to prevent people from escaping there. If I were in charge of a state park, I guess I would be nervous of having all these people "running loose" in the park after some disaster, not knowing how long the situation will last.

As a place to stay put rather than just meet, it may not be the best place for people/families who are not equipped or experienced to live outdoors. Or maybe you're fine, but other people are dumb about using fire and start a wildfire or don't think of sanitation and end up excessively polluting the water for everyone else. And if you get in trouble, like get caught in rain and someone is going into hypothermia, then you're just that much farther from any sort of help than if you were closer to civillization.

Once the authorities are able to get things semi-organized, like send out water tankers and relief supplies, set up aid stations, send police/National Guard troops, etc., state parks probably are not on their list of locations to support unless there are a lot of people there with you.

Another totally random thought that just came to mind--marijuana plots and meth labs. A rare but pervasive problem in some parks. If people start pushing into deeper areas to get away from crowds, then there's the chance of stumbling onto one of these encampments. Even if the occupants aren't around and armed, there could still be booby traps and dangerous chemical contamination.

Anyway, some random thoughts.