Here's a good report on all roads leading into Yellowstone. We've done the Cody, south, west and north entrances with our 40' MH. We have not done the NE entrance with the MH but have driven it.
You are south of the park. Where are you planning to stay? If in Cody then take WY372/28 to Lander and on to Shoshoni then north on 20 to Thermopolis and 120 to Cody.
If staying in West Yellowstone, take 30 to Montpelier and 89 north to Jackson then north into Grand Teton Nat'l Park and to Yellowstone to W. Yellowstone. Or 189 north from Evanston to 89 is an easier drive. If you didn't want to drive through Grand Teton (beautiful) at Alpine,WY on 89 to Jackson/Grand Teton you could pick up 26 west at Alpine to Idaho Falls and 20 north to West Yellowstone.
Do not take Teton Pass (Hwy 22) west from Jackson. It's steep.
Many different ways to enter Yellowstone depending where you want to be. Have a good trip!
Road to Yellowstone
Each national park has a great website giving maps, road information, openings, closings, special alerts (COVID19) and things to do. Be sure to check them out.