How soon can you park

All around the Amphitheater. There are multiple parking places

Very early...park near the exits or--better yet--on the road pointed down to beat the traffic jam on the way out!

I believe it is open dusk to dawn

We just saw Beck there this week , workers encouraged us to come back around 8am to look around

I would say just before sunrise is a safe bet

The parks hours change depending on the season the parks website is likely to be the best place to find current information.

You can park there starting early am.