My wife and I tried to go in August and didn't realize how far we'd have to walk. We missed our tram ride and it started raining (long walk outside from the garage to the entrance and we had no umbrellas) so we decided to cut our losses and leave without really seeing the arch.
Down there. If you want to get in and out you can always park across the river and take the metro over.
https://www.youtube.com/watch?v=dP2lyc53qtI
Not sure what (if any) cost comes from parking there.
It’s been several years since I’ve been to the Arch, so others might know better spots as they’ve done a lot of work around the Arch grounds.
You might get lucky and find a spot at the Landing. Otherwise, it's mostly street parking.