I guess it all depends on what you are considering a "major airport". I think most NP probably have access relatively close.
Yellowstone NP is about 90 miles from Bozeman/Yellowstone IAP, through the Gallatin NF (think Yellowstone without the park service).
Glacier NP is about 25 miles from Glacier IAP (Columbia Falls), through the Flathead NF.
Teton NP is about 5 miles from the Jackson Hole, WY airport.