Nope your not that lucky, drones are prohibited in all national parks in the US. So I wouldn't plan the trip around the idea of just shooting aerial, however, all of the places you mentioned are vast areas, if you use common sense, don't bring out your quad in the presence of other people and start setting it up, especially park rangers thinking that if your nice and explain how safe you are etc - they don't care, and in the Grand Canyon for example don't pull it out in the parking lot and walk to the rim expecting to hike off into more remote areas to launch from and stay away from the touristy areas. Don't hover above the glass lookout platform trying to get shots of your girlfriend. Also, don't pull it out when you first arrive to a location, take your time and watch the flow of people in and out of an area, the flow of people will change depending on tourists busses.
If your shooting aerial at the Grand Canyon don't fly out beyond the rim, if you crash for any reason you need to make sure its in an area that you can safely retrieve it. If your out beyond the rim and crash on the valley floor you won't be able to get to it so at the very least you'll be littering, so not only will you not get your quad back but the fine will probably be more than your quad was worth.
Common sense goes a long way, stay away from people and if your approached by a Ranger be very pleasant and don't speak English.. Plead ignorance.