Fossil collecting is permitted in most Ohio State Parks within designated areas so you would need to check with the Ranger station at the park.
Here is a link to current fossil collecting leaflet from ODNR www.dnr.state.oh.us/Portals/10/pdf/GeoFacts/geof17.pdf
Here is a link to fossil collecting rules at Caesar’s creek: http://napc2009.org/wp-content/uploads/2009/06/caesar-creek-state-park-in-warren-county.pdf
Last time I heard, these permits were still free.
Pick up your permit at the U.S. Army Corps of Engineers Visitor Center, and while you are there check out the center’s display of fossils found at the park. The center is accessible from Clarksville Road and is open from 8 a.m. to 4 p.m. Monday through Friday, and 8:30 a.m. to 5 p.m. Saturday and Sunday. For more information about fossil hunting at Caesar Creek, call 513-897-1050.
Here is a link to current parking restriction regs for Ohio. http://www.dot.state.oh.us/Divisions/HighwayOps/Traffic/miscellaneous/Pages/Regulations-ParkingRestrictions.aspx
I believe in most places common sense still prevails (unless marked otherwise). Be mindful, no parking signs can be small. Last fall, I was the received a $250.00 parking ticket for illegal parking near a hiking/rail road tunnel trail in Wisconsin! A wonderful trip that had a poor and expensive ending.