Ohio Rev. Code § 3111.26
After an acknowledgment of paternity becomes final and enforceable, the child is the child of the man who signed the acknowledgment of paternity, as though born to him in lawful wedlock. If the mother is unmarried, the man who signed the acknowledgment of paternity may file a complaint requesting the granting of reasonable parenting time with the child under section 3109.12 of the Revised Code and the parents of the man who signed the acknowledgment of paternity, any relative of the man who signed the acknowledgment of paternity, the parents of the mother, and any relative of the mother may file a complaint pursuant to that section requesting the granting of reasonable companionship or visitation rights with the child. Once the acknowledgment becomes final the man who signed the acknowledgment of paternity assumes the parental duty of support.
R.C. § 3111.26