I believe it's Old Yellow Stone Trail South. Hard to find good info. on it, but it seems to be part of an old stage coach route. If you look at Google Maps Street View where the north end merges back into U.S. 89, a little ways above where SR 540 branches off from U.S. 89, you can see that it's a dirt road. There is also a portion of it you can cycle in the park between Mamouth Hot Springs and the north entrance. Open one-way to cars but both ways to bikes. Check the park's web site for details.
Could be a nice ride. Last year we did a 20 mile section of unpaved road between Melrose and Twin Bridges, MT that turned out to be a hidden gem. If it's rough going, you can cross back to U.S. 89 at Corwin Springs and Miner.