As it turns out there has been a little bit of an issue with HUNDREDS of cars being parked outside the park. Many illegally. Now they have sigs and are enforcing the rules. I doubt you could predict where an open spot would be so I think just drive to the gate then make a u-turn and art in the first legal spot you see. I hear there are some signs now pointing to parking after residents complained about people larking in their yards and driveways stuff like that.
Hikers care more about being close but if you have a bike an extra four miles is not a big deal.
People are camping on the BLM land adjacent to the park because they can drive a car there and "tailgate camp". But you can still camp legally in the park so long as you are one mile from any road.