Keep driving
The pin drops in the area with “no camping” signs but if you keep driving up the road past the stables, there’s plenty of spaces. Beautiful in the snow, and it’s a well graded road, you don’t need 4 wheel drive.
Perfect spot if you don’t want to pay to stay in Yellowstone
15 minutes from the Mammoth Hotsprings visitor center. Easy to access, no crazy potholes or anything, just a dirt road. There’s a couple cabins up that road too so I assume whomever owns them keeps the road well maintained. The first spot you come to has an incredible view which is where we stayed, but if you want something more sheltered keep going around the bend and there’s a little rocky outcropping/quarry type area. It was absolutely freezing though, the inside of the truck bed had frost in the morning (22°!)