At handgunlaw.us/states/ohio.pdf, there's a reference to an Ohio Supreme Court ruling in 2008 on this exact question (Slip Opinion 2008-Ohio-4605). Basically, it says that while municipalities are lawfully able to create ordinances in their interest, conflicts with the state's preemption over the field of arms-related legislation reigns supreme. Hence, carry is lawful in city parks.
The issue presented in this case concerns whether Clyde Ordinance 2004-41, which prohibits licensed handgun owners from carrying concealed handguns in Clyde city parks, is a valid exercise of the municipality’s home-rule power according to Section 3, Article XV III, of the Ohio Constitution. Because the ordinance is an exercise of the municipality’s police power that conflicts with a general law, the ordinance is unconstitutional. Accordingly, we affirm the judgment of the court of appeals.