Starting on 13 June 2022 and lasting for an unspecified / indeterminate period, Yellowstone National Park is not allowing new visitors to enter due to severe flooding and consequent road damage at various locations throughout the Park. Refer to the Park's "Current Conditions" page for the most up-to-date information and conditions.
New release: UPDATE: All entrances to Yellowstone National Park CLOSED temporarily due to heavy flooding, rockslides, extremely hazardous conditions - Yellowstone National Park (U.S. National Park Service) (nps.gov)
Current Conditions: Current Conditions - Yellowstone National Park (U.S. National Park Service) (nps.gov)
Park roads map (interactive): Park Roads - Yellowstone National Park (U.S. National Park Service) (nps.gov)
In case you missed it elsewhere, the southern portion of Yellowstone has reopened in a limited fashion. Check the NPS Yellowstone website for the latest status information.
Please be aware that many of the campgrounds that touring cyclists use remain closed. Which makes it very difficult to bike through the park. At present only Grant Village and Bridge Bay are open. Plus, the USFS campgrounds just outside the park are often filled. Private campgrounds charge huge rates. When all campgrounds are open you can ride moderate distances at lowere traffic times. Having to ride considerable distances puts you riding at the very busiest times with extremely high traffic volume on the pakr roads.
Plus, there are reports of extremely long lines to get in. I do not know whether they let cyclists go ahead, but people waiting hours may not take kindly to that.