Ohio started keeping birth and death records on the county level in 1867, although a few counties have records for 1856 (not Jackson). These are found in the probate court of each county until 20 Dec 1908, when they then became a state record. After that date, they are kept in each county's health department, as well as at the state Health Dept. Birth records have always included the name of the parents, although not always the maiden name of the mother. Death records prior to 20 Dec 1908 were only required to have the names of parents for infants, but some clerks at some times included the names of parents of adults as well. Good luck!