Most, (not all, but most) public parks in the US have posted hours, usually dawn to dusk.
Sometimes they are more specific, such as 5a.m. to 9p.m.
The park by my house (Southern US) is open from 6a.m. to 10p.m. Sometimes me and the wife will go there after closing, because we work nights. The cop that works that patrol stopped us once for being there so late, but after he saw that we weren't hurting anything, he left us alone.
After these hours, you are trespassing. Depends on the location/cop if he wants to bug someone who is just sleeping.