Ohio State law/Statutes trumps all City and County rules/regulations... There is another way to say that, but I cant think of it at the moment....
In OH, if you interact with a cop for "official" purposes, and have a permit AND you are armed at the time, you need to notify the cop that you 1. have a permit to carry, and 2, that you are currently armed, I suggest strongly doing it in that order! lol
You can carry concealed in any city or county park..... However, if there is a building there that is normally occupied during working hrs by staff, it is more than likely (read IS) off limits to carrying a firearm in. That would be something like a park rangers office, places like that.... but restrooms, picnic shelters and such are ok to carry in.
Also stay away from School zones........ and anywhere that sells alcohol for consumption on the premises.., the law fixing that still isnt in affect..., so no restaurant carry if they sell alcohol!
Have fun in Ohio, although I am not sure that will be possible in Cincinnati.....