Travel within the Yellowstone and GTNP can be dicey that time of year so you will need to take into consideration road closures. I also believe that there are already some road closures within Yellowstone due to construction.
My girlfriend and I went to Yellowstone and GTNP the latter part of October 2020. We drove up from east Texas since we both enjoy driving/riding and seeing the countryside along the way. We entered the park in the Northeast Entrance and stayed at Mammoth Hot Springs Hotel (no TV's) for several days while we explored the upper central and northern part of the park hitting the usual sites.
We then transitioned to Jackson and stayed at the Rustic Inn Creekside Cabins on the north side of town. They are really nice with fire pits around the property and teepees for the kids to play in. Moving to this location allowed us to explore the southern part of Yellowstone and the GNP with ease.
We did more hiking in GTNP than in Yellowstone. For me, Yellowstone has more "site seeing" stops that you get out and walk around and explore whereas GTNP requires getting out and doing some hikes around the lakes, etc. There are hikes in both parks that are suitable. Just carry bear spray.
We were there between the "prime seasons" so the lodging rates were lower. Plus the crowds were not so big either. Downside of going out-of-season, coupled with Covid, is that some amenities are closed. Food service in the parks is limited.
There are several nice restaurants in Jackson so there are no issues in finding food for the evening. For food going into the parks we stopped at the Creekside Market and Deli each morning to pick up sandwiches for lunch. They are fantastic.
Keep checking local conditions for that time of year to help plan on how to visit all of the parks.