If I wanted to film a scene, when would you suggest is a best time that won't disturb tourism? Also where to find permits?

Honestly we went on a Sunday at around 4-5 pm and there weren't too many people there.

I agree...

Get a permit from the park service. A lot of people go in the morning and afternoon. It's always packed but weekends have more people and Tuesdays would have fewer people

You need permission from the Park Service for a commercial shoot.

It's always busy, and packed morning best time

First thing in the morning on a weekday is probably the least busy.