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.
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Currently, the use of drones is prohibited in Grand Canyon and all National Parks.
No drones in Yellowstone National Park.This means that area visitors are prohibited from using drones once they enter the park gate. The U.S. Forest Service prohibits drone use in wilderness areas, but allows drone use on general forest lands.
Unfortunately, you cannot put your plan into action. National Parks were classified as “no drone” zones in 2014. And flying here means stiff penalties. For drone pilots, this is a huge missed opportunity — the NPS controls some 417 parks, 23 trails, and 60 rivers.
If you fly a drone in an NPS-administered park, you risk interfering with the flora and fauna of the location. You can also disturb park visitors, hikers, bikers, camping tourists, etc. Drone pilots caught breaking the rules will be fined $5,000 and risk up to six months in jail.
Use of remote-controlled equipment (including but not limited to helicopters, drones, and other aircraft-based equipment) within Zion National Park is prohibited. For more information visit the NPS Unmanned Aircraft webpage.
For a safe and enjoyable visit, please know these park regulations and safety advisories. For more regulations see Grand Teton Superintendents Compendium and Code of Federal Regualtions above. Launching, landing, or operating an unmanned aircraft, such as a drone, is prohibited.