Re: your dogs.
In Zion, the only way to see the canyon is to take the park shuttle bus. Nice system, but no dogs allowed. Now if you get electric hookups, you could leave your MH plugged in with the A/C running while you go away for a few hours. Even if you stay someplace in town, there is a shuttle bus that runs into Springdale that will bring you into the park. Otherwise, someone will have stay with the dogs or you might board them for the day in town (research this and make reservations in advance if possible).
Grand Canyon is a little more pet friendly along the rim, but the dogs aren't allowed on the park buses, which is the only way to see the whole section out to Hermits Rest (unless you want to hike). Plus, access to much of the main developed south rim area is by shuttle bus or by foot from your vehicle, and since you don't have a toad, you'll find parking for RV's much more limited. (Yes, the parents will be walking, but it's mostly paved trails and sidewalks with plenty of places to sit and rest.) Maybe you can drop the parents and dogs off near the lodges and then go park and meet up with them. There is a kennel at the GC if that becomes easier.
Bryce, Arches, Canyonlands, Yellowstone are all very restrictive on pets. In all of these parks, you really need to get out of your vehicle to see the main sights. For instance at Yellowstone, you have to park and walk just to see Old Faithful.
I guess what I'm saying is that you really need to pre-plan you vacation around the limitations imposed by the dogs. We have it easy with the "camping cats", as they have no desire to go outside and they have their litter box right in the camper. And there are still trips we take to Utah where it will be too hot for them and we leave them at home and pay for cat care. I know pets can really be a part of one's family, but is it possible that it will be easier on them and you if you have another family member take care of them for your trip(maybe your brother in TN)? Otherwise, I'd suggest researching each park's regulations and plan how to sightsee in each park so everyone gets to experience them.
