The Ohio State Fair started yesterday. The Ohio Historical Society is located next to the state fairgrounds and is used for parking during the run of the fair. If you are going to visit the Historical Center or the Library/Archives during the fair (which runs through August 10), you will need to pay $5 for parking.
If you are a member of the Ohio Historical Society, be certain to bring your membership card with you! If you show it at the gate, you will get free parking.
Presuming you get through the traffic on I-71, State Fair time is actually a good time to visit the Library/Archives at OHS. The fair crowds tend to scare off would-be researchers, so the library is usually pretty empty.