The best bet is at Hollywood Studios about an hour before park closing on nights without Fantasmic. I've driven in even earlier than that through the front entrance and the attendants were gone.
I think it is a safe bet for sure after 6 or 7. I think it depends on how late the park is open, how many cars are coming in etc. so there isn't an exact time.
There is a guest relations at Down Town Disney. In fact, that's where the head GR offices are, if you can take care of your business there.
I will say that the parking is transferable from park to park (park at EPCOT, drive to DHS, pay once).
While this is very subjective, the parking rate at Disney is not all that terrible. In fact, it could be far worse
I didn't know about the Downtown Disney guest relations. Since we are staying about 2 minutes from there, I will make that my first stop!
Thanks for your help!
Walt Disney World's Magic Kingdom
Disney's Animal Kingdom
Disney's Hollywood Studios
Disney California Adventure
Universal's Islands of Adventure
Universal Studios Florida
Universal Studios Hollywood