Near Bryce Canyon, we stayed in Panguitch at the Hitch-n-Post. It's right off the main drag, but wasn't noisy. Nice owners that live on site. About a 20 minute drive to the park.
For Zion we stayed at Bauer's Canyon Ranch RV Park. A little further, 1/2 hour to the park, but a nice drive and you get to see the east side that lots miss. You can also go north from there and across UT14 to Cedar Breaks and down into Cedar City. Once in Cedar City, go south and visit the Kolob Canyons area of Zion, and then down to the main park. Nice RV park, not too spectacular, but we'd go back.
In Moab we stayed at the OK RV park and stables. It's a longer drive, almost an hour up to Canyonlands and 1/2 hour to Arches. It was a very nice park, the roads were gravel, so it might be dusty later in the year. It was the only park we saw that was not right on US191 which is a major truck route. I could only imagine the road noise at the others. OK didn't have any of it.
We went to these in April of 2012 for Moab and 2013 for Bryce and Zion. Both seemed pretty popular, so I would recommend reservations for all 3. In Moab we had a 40ft 5er. We traded that in and brought our 40ft Class A to Bryce and Zion.