The FAA has authority over almost all of national airspace, which includes the airspace directly over National Parks. However, the National Parks Service (NPS) has the authority to restrict flight take off and landing from its property.
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The purple zones on the FAA chart are no-fly zones. But there are corridors where you can fly between the zones—the Tuckup, Fossil Canyon, Dragon, and Zuni Point corridors—which just happen to be the most spectacular parts of the Grand Canyon. They did that on purpose, to allow pilots to fly over this great stuff.”
You'll need permission and licence when using drones above Natural England's land to avoid disturbing wildlife. Drones should not be flown on or over Sites of Special Scientific Interest (SSSI) or Special Protection Areas (SPAs) without consent from Natural England.
You can not fly with any UAS in a national park unless you got your P.107/Drone Pilot license, a permit and a waiver from the FAA. Other than that, without the waiver and permit you can NOT fly in a national park at all.
Recreational and sport pilots can carry only one passenger at a time; private pilots may carry as many passengers as the airplane will legally seat, provided it stays within the airplane's performance and weight-and-balance limitations.