One of the best—albeit not the warmest—times of year to bicycle in Yellowstone and Grand Teton national parks is during the month of April.
As you can see by clicking on this link, bicycling and other means of non-motorized travel—in-line skating, walking, etc.—may be enjoyed in the world's first national park on the roads between West Yellowstone and Mammoth Hot Springs in April. The South Entrance road and part of the East Entrance road are also open to bicycles (while closed to cars), as conditions allow. The dates depend on the severity of the preceding winter and other factors; or, as the National Park Service puts it, “The first day of ‘spring bicycling’ is never predetermined and is dependent on road conditions as determined by park staff.”
To the south of Yellowstone, in Jackson Hole, Wyoming, the Teton Park Road becomes one of the world’s great bike paths in April (the road opens to motor traffic on May 1). Serious plowing began this year on March 24, with rotary plows clearing the way between Moose Junction and Jackson Lake Junction, a distance of about 20 miles … 20 miles of smooth, traffic-free pavement serving up some of the most spectacular mountain views on Earth. After this past winter’s prolific snowfall, the banks will be high, but the road will be dry (for the most part).
Please note this caveat from the Park Service: “Although the Teton Park Road will open to non-motorized use, visitors should be alert for park vehicles that may occasionally travel the road for administrative purposes and for snow plowing operations that continue as a result of recurring snowstorms. … As a reminder, entrance stations are operating and collecting fees.”
Come prepared to bundle up, although there's always the chance of hitting it on an unusually warm and sunny, western Wyoming spring day. If so, enjoy!