I see your points. I don't travel cross country so I have no experience in this. I just know that these city owned parks exist. It may not always be convenient to use them.
A lot of them are self check in where you just pick a spot, fill out a envelope, and then put it in a slot with the money in it. But, almost all of them are a no noise after 11pm sort of thing so stopping at 1:36am would not work.
I guess I was think of traveling with a plan, knowing the route in advance, and then having phone numbers handy of the parks along the way so you can make a reservation on the go. I understand that for some people it doesn't work that way.
Forgive me I'm my dad's son. When we took family vacations as a kid we knew exactly how far we were going to go and where we were going to stop. Everything was planned out to the minute.