You cannot drive to most of the major parks on one weeks time. If you plan to fly in then the major parks are Yosemite, Yellowstone, Tetons, Glacier, Bryce Canyon, Zion, Grand Canyon, Capital Reef, Lassen Volcanic and maybe Olympic. On the east of the Mississippi side you have Great Smokeys, Arcadia, Cape Hatteras, Everglades (best in Fall/Winter) and Shenandoah. Add Blue Ridge as a nice several day drive.
There are a lot of great state parks too. Florida is packed with them as is the area around the great lakes.
With a six year old (been there three times) I'd stick to the eastern parks. By the time you've done them he'll be better able to keep up and maybe hubby will get more vacation time. In my experience six year old kids are on the edge for trails, having one pull a sit down strike 2 or 3 miles from a trailhead, forcing you to carry him that far will quickly sour your day. We had that happen at GSMNP on the bald head trail.