Hmmm ... the problem with doing a "national parks" tour out West is the distances are so far between most of them, and the towns and cities are spaced much farther apart than in the East. However, there are few NPs that are close enough together where you could combine them in a trip ranging from 200-300 miles. Here are a few that come to mind:

- Zion, Bryce Canyon and Cedar Breaks in SW Utah, perhaps combined with the north rim of the Grand Canyon.
- Arches, Canyonlands and Capitol Reef also in Utah.
- Redwoods, Lassen Volcano, Lava Beds and Mt. Shasta in N. California, perhaps combined with Crater Lake, OR.
- Yellowstone, Grand Tetons in WY and Craters of the Moon, ID.
- Joshua Tree, Mojave and Death Valley CA.
- Badlands, Mount Rushmore and Black Hills, SD, and Devil's Tower, WY

I could suggest others but these are all parks that I have personally visited. You would have to check actual distances between parks to determine what is realistic on bicycles with kids. A few comments about these areas. Traffic can be very heavy, like rush hour in a big city, in Yellowstone and other very popular parks. In contrast, the NPS highly restricts car traffic in Zion, making it surprisingly bicycle friendly. By car, Zion, Grand Canyon N. Rim and Bryce are within easy driving distance and all very scenic in their own ways.

We combined a trip to Redwoods NP and Crater Lake a few summers ago (driving a car), and it was fantastic. Two very different parks that are equally scenic in different ways. It was almost like having two vacations for the price of one. The drive between them was also very scenic, particularly the roads in southern Oregon leading to Crater Lake.

The desert trio in Southern CA would be fantastic during the right seasons but would be too hot in summer. We visited Joshua in spring while the cacti were flowering and it was spectacular. I haven't been to Death Valley but apparently it is incredible if you time it during wild flower season, assuming there's been enough rain for flowers to bloom.

The Badlands-Black Hills-Devil's Tower area is also worth consideration. We saw more wildlife in the Black Hills than anywhere else in the US we have visited. Devil's Tower is just cool. Badlands surprisingly scenic but I wouldn't want to be there in hot weather.