We just recently went and we got there around 7am on a Sunday and didn't have an issue parking in the park. We also went back during the day (to stop by the visitor center) and there was no parking (12pm). Definitely buy a shuttle pass ahead of time though!
The Watchmen trail is one of the few trails you don't need the shuttle bus for. You can walk into that trail. If you are doing Scout's Lookout, Angel's Landing or Narrows or any trails around those you need a shuttle pass. They open the passes at 9AM Springdale time for the next day. They sell out very quickly, so be sure to login right at 9.
They limit the amount of cars in the park, but there is a parking lot at the visitors center and you can take the shuttles in. They are easy to use and run continuously. But yes, you need to reserve shuttle spots
You can drive in but parking is very limited
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