I think his parents were:
John B. Gibson, b. 1791 Franklin Co. PA & died 1870 in Blacklick, Indiana Co., PA
Susan Gibson [don't know if this was a mistake on the family tree or if her maiden name was Gibson, b. 1798, PA & d. 1880 Buffalo, Butler, PA
To prove they were his parents, look up John's probate records in Indiana County, PA.
You won't find a birth record for Joseph in Ohio because they weren't issuing them that early.
Married: 31 Aug 1848, Franklin Co. IN
Source: IN Select Marriages Index 1748-1993
I'd look for their actual marriage record and look for parents.
IF they'not on there, then look in the local newspaper for their marriage announcement.
1860 Fairfax, Linn, IA
Joseph Fibson, b. IN, 36, farmer
Nancy, b. IN, 34
Isaac, b. IN, 6
Elijah C., b. IA, 11/12ths
1880 Eudora, Douglas, KS
Joseph Gibson, b. OH, 53, [parents born in PA], farmer
Nancy, b. IN, 48
Riley S., b. IA, 13, works on farm
Isaac L., b. IN, 26, farmer
M. Isabel, daulaw, b. iN, 25
Ora, grandson, b. IA, 3
Josephine, granddau, b. IA, 2
I don't think you'll find a death record for Joseph. I'd look up the Washington County Kansas genealogical and historical society and ask if they have index newspaper death notices. His parents names might be in it. Successful Searching!