Is there a place at the bus terminal where I can park my car for the better part of a day while on a round trip bus trip?

I'd call and ask as there is a lot of construction in the area. If no parking is available at the station itself, parking lots and garages are readily available in the vicinity. If you get there reasonably ahead of departure, you will be able to find a spot without much hassle.
Greyhound Telephone Numbers
Main: (513) 352-6012
Baggage: (513) 352-6030
Greyhound Package Express: (513) 352-6030
Customer Service: (513) 352-6012
The map made it look as if the bus terminal was in the middle of nowhere. Glad to hear about the proximity of parking lots and garages. I will be coming from Bellevue, KY and will check about construction. Thanks for all.
One idea is to park your car in mt Adams and walk across the pedestrian bridge over I-71 that ends directly in front of the bus station.