You never mentioned what your pulling. I live in Michigan and many of our state parks are full this time of year and tight to drive though. People park very close to the road sometimes making it hard to pass by them much less maneuver around. It could be very dark.
Why not drive up close to where your going and find a rest stop or parking lot that would be much easier to stay the night. You won't worry about fighting in the dark to squeeze in somewhere. You might be asking someone to move to allow you to get in, they could be sleeping. Or leave early in the morning to put you there when the site opens. Check in is probably after 1:00 anyways.