No, it is a STATE park, not National. If you go to the aerial tram you will have to pay for parking because it's not part of the park. I believe parking anywhere else is free, Idyllwild, Palm Springs. Lots of hiking.
Check with the local ranger station in Idyllwild there because when I park at Mount San Jacinto I typically go there just to pick up a sheet for the number of people that are in the party and we carry that sheet with us in case something happens to one of us or all of us. I also have a parking permit for the Los Angeles Forest but I haven't used it or needed to use it for Mount San Jacinto.
No
I don't think so
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