I assume that you and/or your Dad have a Senior Pass so you won't have to pay the entrance fee just to drive through Yellowstone.
Check the road construction in Yellowstone, as that may affect whether you want to drive around the lower portion of the lower loop or the upper portion to get to Fishing Bridge and the East Entrance. It's all downhill from Sylvan pass and the section inside the park will require lower gears. Once past the entrance station the grade decreases.
Most folks take suggest route 16 over the Bighorns for lower grades, but we prefer route 14 for the scenery. Lower gears needed on either going up or down.
Lots of folks in RVs do all of the above, and since you indicate past experience, you should have no issues.
2003 Winnebago Minnie 24F - Ford E-450🙂