Just in case you haven't found the rules
Here you go. http://stonemountainpark.com/faq/
Do you have bike trails or somewhere for me to ride my bicycle at the Park?
Bicycles can be ridden along the roadways inside the Park. There is a dedicated bicycle lane on Robert E. Lee Blvd., on the back side of the mountain. Bikers must ride single file and follow all traffic laws.
I'm guessing that since the park is owned by the State of Georgia, state traffic laws apply and that there is traffic through out the hours of operation.
What time do the Park Gates open? What time do the attractions open? Park Gates are open daily, 6:00 a.m. to Midnight. Attraction hours vary. Visit the Operating Dates & Hours page for attraction hours on the day of your visit.