I live fairly close to Zion and do multiple day trips each year. Pa'rus was being worked on a lot last year and that eliminated some of our enjoyment with the pups.
We stayed in Watchman for the first time with our TT in November and naturally ALL the campgrounds are dog-friendly for walking polite dogs. We had no problem creating multi-mile routes for ourselves.
You can also carry your pass and walk out of the park and into the town of Springdale, where you will find multiple restaurants which allow dogs on the patio dining areas (one even had us walk ours through the restaurant to get to the patio).
We haven't gone to Bryce yet with the dogs, but I second the suggestion for Ruby's - just ask to be put in the area with more trees if the weather might be warm.
Also remember that a lot of Utah that is not National Park land is BLM land which is almost always very dog friendly.