Yes, I visited Bryce a long time ago, and we ended up at Red Canyon state park just down the road. There was nobody there (it was spring and there was still some snow) and it was beautiful. A good rule of thumb is that BLM and National Forest are usually very dog friendly. You can also camp for free for up to two weeks in any non-posted spots in the National Forest (we took full advantage of that when working the summer season and saved on rent...). I had my dog and we never had any trouble.
The one thing I always have to add is that my dog did end up with the fungal infection Valley Fever after that summer- he probably got it on road trips to Arizona, as I don't think it's in Utah, but do look into it and make sure you do your best to avoid exposing your dog.
Enjoy! I'd love to go back someday.
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