Capitol Reef National Park -- Drive time: 3 hr 57min. A bit hot this time of year, but you could spend the afternoon hours by the pool in Torrey and the mornings and evening hiking. Darkest night sky of any Park in the US; check out their star parties. El Diablo Cafe is a gem.
Zion National Park -- Drive time: 5hr 4min. Busy this time of year, but the Kolob section would be less over-run than the main canyon. Relatively cool cause of it's elevation. Also, plenty of trails that are easily accessible by the Park-run buses. Do Angels Landing. Check out Cafe Oscars in Springdale (which offers grass-fed beef).
Canyonlands National Park -- Drive time: 4hr 49min. Plenty hot, but you'll have the place to yourself. Rent a jeep in Moab and take a couple days along the ~100mi White Rim trail. Let me know how it is. One of the most desolate National Parks.
Arches National Park -- Drive time: 4hr 40min. Delicate Arch will be crowded, but there are plenty of lesser-known hikes and you can always find stuff to do around Moab (bike slickrock, run the river, jeep, etc.).
Great Basin National Park -- Drive time: 5hr 1min. One of the least visited National Parks. Take your snow shoes, if you wanna do Wheeler Peak (13,000+ feet). There's a cool cave and plenty of open space to camp without being bothered. Also has star parties and super-dark night skies. The primitive campsites are also a great option, giving you a toilet and fire pit without the crowds. Cooler weather also.
Bryce Canyon National Park -- Drive time: 4 hr 43min. Probably would provide a good break from the heat this time of year. Get a permit to camp in the backcountry and you could probably find miles of trail to yourself. It's amazing how few people actually hike in these Parks. The hoodoos are a must-see for any American. The trails in and around the Amphitheater provide an amazing amount of "oohs" and "aahs."
Grand Teton National Park -- Drive time: 5hr 43min. Cooler weather up here also, along with the posh town of Jackson Hole. Run the Snake River, hike to Jenny Lake, shop and dine in Jackson.
Yellowstone National Park -- Drive time: 6hr 34min. Also cooler, but sure to be bustling this time of year. It's a big park, however, so you're sure to find some solitude if you're willing to hike. See Old Faithful, watch rare wildlife, camp with nothing but what's on your back in the nation's first National Park.
I'm always amazed by how many people come from around the world to enjoy what's in our backyard; such accessible empty, stunning landscapes are found in very few other places. Let's enjoy, support, and preserve one of the amazing things that America's leaders got right.
Stay safe out there.