Preliminary Thoughts

"Where does one cycle once one gets there?" I've never had the pleasure of visiting the place so can't personally speak to the conditions but Google streeting what is available does not look good and the National Park Service Website is not encouraging.

Biking Yellowstone
"Use extreme caution when riding on park roads; roads are winding and narrow while shoulders are either narrow or nonexistent. Vehicle traffic is heavy most of the time. There are no bicycle paths along roadways.

We strongly recommend that safety gear, including helmet and high visibility clothing, be worn by all bicyclists. During April, May, and June, high snowbanks make travel more dangerous. Road elevations range from 5,300 to 8,860 feet (1615 to 2700 meters), and services and facilities are relatively far apart--typically 20 to 30 miles (37 to 56 kilometers).
Motorists frequently do not see bicyclists or fail to give them sufficient space on the road. Drivers sometimes pass on hill crests, blind curves, or in oncoming traffic. Vehicles, especially motor homes or those towing trailers, may have wide mirrors, posing an additional hazard. For more information about bicycling, stop at a visitor center."

. . Perhaps that was written by one of the proponents of totally banning cyclists from National Parks except for dedicated bike paths but the terror of the breeze from rear view mirrors passing within inches of my left cheek trumps any and all other considerations.
Bruce