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If you arrive early enough you can park at the visitor center. But it is often full before 10 am. So, if it is full, you'll have to backtrack into town:
The National Park Service has a map of park & ride areas in Springdale which you may be able to use. Some of these are standalone park and ride facilities; others are sections of hotel or local business parking lots, or adjacent to them. In these cases be sure to follow the Park & Ride signs (which unfortunately are fairly small and can be hard to spot) and park in the marked areas. This map also shows where the town shuttle stops.
The Park & Ride sign looks like this:
There is also street parking on parts of Zion Park Blvd., but don't count on finding an open space.