That is how I get by, flying the park perimeter.
I have planned a few Litchi missions for just planning sake - may never get there.
But I found you still have to look deeper into the surrounding laws/ regulations.
Case in point, Devil's Tower National Park Wyoming.
Easy for a Mavic Air to get all the way around the park staying outside the 6 mile (9.65km) or so perimeter.
But in digging deeper, I discovered a recently constructed 2003 airport to the north east - Hulett, Wy (W43) has an agreement between the park and itself.
NFZ 2 Nautical Miles from the center of the tower year round except June
NFZ 3 Nautical Miles from the center of the tower throughout June
https://www.nps.gov/deto/planyourvisit/upload/DETO-No-Fly-Advisory-Bulletin-Final.pdf
I can only speak for Texas State Parks and they have left it to the individual park superintendent.
I also found the superintendent can help from time to time.
When I asked to fly, he said he would love to let me fly but can't for 2 reasons.
1) The park is within 4.9 miles of the local airport (seems we were both uneducated at the time, 2016) and
2) The park is a Wild Turkey Refuge.
But then he went onto say "But if you go outside the gate and travel north on 377, there is a scenic overlook of the park that is 5 miles away from the local airport and is outside the park."
So you see, it all depends.
If you do not ask, you will never know.