Acadia
American Samoa
Arches
Badlands
Big Bend
Biscayne
Black Canyon of the Gunnison
Bryce Canyon
Canyonlands
Capitol Reef
x Carlsbad Caverns
Channel Islands
Congaree
Crater Lake
Cuyahoga Valley
Death Valley
Denali
Dry Tortugas
Everglades
Gates of the Arctic
Glacier
Glacier Bay
x Grand Canyon
Grand Teton
Great Basin
x Great Smoky Mountains
Guadalupe Mountains
x Haleakala
Hawaii Volcanoes
Hot Springs
x Isle Royale
Joshua Tree
Katmai
Kenai Fjords
Kings Canyon
Kobuk Valley
Lake Clark
Lassen Volcanic
x Mammoth Cave
Mesa Verde
x Mount Rainier
North Cascades
x Olympic
x Petrified Forest
Redwood
? Rocky Mountain
Saguaro
Sequoia
Shenandoah
Theodore Roosevelt
Virgin Islands
Voyageurs
Wind Cave
Wrangell-St. Elias
Yellowstone
Yosemite
x Zion

Been to the Rocky mountains, but I’m not sure it was the National park.
Rushmore is a monument, not a park.