Free Alternative Outside the Park
Hard to find and notice on the side of the road, I drove past it a couple times after I couldn't find an open site in the park. It's very bumpy but you don't need 4wd. Not super pretty and you might struggle to find spots flat enough for a tent, but at least it's free! Although, there are lots of shattered bottles in certain spots so be wary.
I slept in my van both nights and it worked out great as we were like 5 minutes from the park entrance; worth it if you're sleeping in your car!
I believe you'll find bigger and more comfortable spots further West though, such as:
Month of VisitApril