Sorry - a few months late to the conversation but worth adding some knowledge to this thread. You will often see comments that you need a permit to shoot video of a national park commercially., or "for money".
First, let me address "for money", the rules are generally "commercially" which is different than "for money", or "for profit". I hardly ever sell my drone footage, most of it is just "for me" - but I am a commercial photographer, videographer, designer. As such any posting of video that might promote my work - is "commercial". I don't have to try and sell it - it promotes my commercial work and therefore is commercial.
Now on to shooting commercially in a National Park. As long as you are not using: crew, props, lighting, etc - it's not needed. Call the park ahead, be kind, let em know, they will tell you - thanks for letting us know - you don't need a permit, have a nice time at the park. When you are obtaining permits its because your commercial work will impact the park more than normal and you are likely to need oversight from park staff to manage that impact.
There are numerous articles on it. Just google national park video film permit.
Last Point: Wilderness / Preserve areas
The rule from the FAA AIM manual is flight over noise restricted and wilderness designated areas has a floor of 2,000 ft. AGL to the highest object within blah blah blah - you can look it up and read the whole thing - you got the big number though, right? 2,000 ft.... you can't go there so you can't go over. Not without a waiver. If you read your sectional you'll see the boundaries and notes are on the flip or in a margin that will remind you - no. Now you are welcome to launch and land from legal land outside the area and shoot towards it within airspace that you are permitted to fly from. That's totally fine.
You might say - but I see helicopters do that all the time. Yup, they do. Some are breaking the "rule" some have waivers.... You could technically break the "rule" as well - get in some trouble make bad press and contribute to ruining it for the rest of us - please don't. 🙂
Hope that helps!