In my experience National Parks are very difficult without a car. The ones that I know of that have shuttle systems are Grand Canyon (South Rim) and Zion BUT you'd need to get there and you'd need a car for that in my view. I believe Lola knows a good deal about Grand Canyon.
Your dates are very early in the season for any of the parks in the Northern Rockies. Yellowstone essentially opens Memorial Day weekend but you'd need a car as there is no public transport in the park nor to the park. Glacier's main scenic road thru opens at the earliest at the end of June. Grand Tetons is smaller than either of them but still no public transport or shuttle on the park roads.
IF you were willing to drive and do a road trip you could go in/out of Denver and make a loop up to the Black Hills (Mt Rushmore, Custer State Park, Badlands, Minuteman Missile silos), head across to Cody WY picking up Devils Tower and perhaps Little Bighorn Battlefield, then Cody for a night or 2 (excellent 5-museum Complex). By that time it might be close to Memorial Day so you could so a minimum of 4 nights in Yellowstone, 2 or 3 in Grand Tetons (more is better for both), then head back down to Denver. This would be a lot of driving but I like road trips so it would work for me. It might not appeal to you at all.
Driving you could also do the "Big 5" or Utah Grand Circle - Arches, Canyonlands, Capitol Reef, Bryce and Zion. This is a good time of year for these parks as some of them can get pretty hot in summer.
