This is a complicated question, and there is no simple answer.
Some towns explicitly allow camping in their city park, and some explicitly prohibit it. Most towns, however, neither explicitly allow nor explicitly permit it. What do you do in those cases? It depends on your risk tolerance. In many cases, if you were to ask the local police, they would probably say "no", but if you simply camp there without asking, they probably won't interrupt you. But note the "probably" words in that sentence. This is where the risk comes in.
One advantage of following an ACA route is that in most towns, they have already investigated and negotiated this for you. If you are on some random route, the towns may not see many bicycle tourists, and may not have even thought about whether to allow or prohibit a bicycle tourist to spend the night in their town park.
I think most of us have done a little bit of everything conceivable in these cases. If there is no explicit sign prohibiting camping, I've just camped. In all cases but one, it's gone off without a hitch. In one town, a police officer woke me up in the middle of the night, but after determining that I wasn't a threat, and that running me off at 2:30 in the morning wasn't really feasible, permitted me to finish out the night. I've even camped on the back lawn of the police station before without permission. That worked well too.
In a pinch, I have camped in places where it was explicitly prohibited. I don't like to do that of course, but sometimes it's getting dark and you have no alternatives. In those cases, it's best to set up after dark, stay out of sight as much as possible, and leave at first light. But you might not get a very restful night's sleep.
To hedge the risk a bit, you should research all the known legal places to camp along your route. In places where there is no known legal place, you could call the local police, sheriff, fire stations and churches. Rather than ask where you can "camp" (which might bring up images of a big deal with a campfire), you might just ask if they know of any place where you can "pitch a tent" or "sleep on the floor" for the night. Make it clear that you will only be there one night and don't plan to take up residence there. Of course you are going to need a bathroom too, so ask if they know of a bathroom that will be open overnight (which often results in them deciding to leave the park bathroom unlocked for that night). I've even attached a note to the bathroom door asking if they could leave this unlocked tonight.
Good luck!