@Crowderdl welcome to the forum and the hobby from another western NCer
In regards to "Getting Permission" from the NPS you're probably not going to want to hold your breath for it here locally. This district is notorious for not giving UAS authorization for any NPS managed land unless it's an ACTIVE Search & Rescue Operation. We aren't even allowed to practice our SAR Operations from NPS land. Key words being "From the land". If you can find PRIVATE property, get permission to be on that property, and fly from/to that property you can fly over "regular" NPS land if no other restrictions are in place (Wilderness Area, Controlled/Restricted Airspace etc).
Now National Forest Service (NFS) land is a whole other thing. You can fly from that area if there are no other restrictions but you'll want to research each and every area you're interested in well in advance.
On the topic of NPS land, it's very possible your DJI aircraft will know that the area is restricted and not even engage the motors due to DJI's GeoFencing/GeoZone restrictions.
Good luck and Safe Flights! We have a lot of beautiful areas here in western NC to explore.
Allen
