You do not need to visit every single of the parks.
You just need to enjoy the ones you visited.
The top spots I would like to recommended are:
1.Yellowstone/Grand Teton, WY/MT/ND
2.Yosemite/Kings canyon & Sequoia, CA
3.Grand Canyon/Petrified Tree, AZ
4.Arches/Canyonlands/Capitol Reef, UT
5.Zion/Bryce canyon, UT
6.Mesa Verde and four corners (including all the monuments nearby), CO/NW/AZ/UT.