Colleen and I worked on this family together. I thought I would update, since Roger was kind enough to reply Colleen's query. We posted this back before we were familiar with the records.
Roger, you are correct. James Rhatigan and Bridget Clyne are the correct parents for John Rhatigan. Bridget must have also gone by Mary, because John's death record gives his mother as "Mary Kline" (Clyne), but a marriage reord for him in the U.S. shows "Bridget Cline." They lived in Carrickedmond. I cannot find a marriage record for James and Bridget, but I found a dispensation in parish records in 1955.
We found a Baptism records for John (1963), Mary (1966), and Patrick (1869), and civil birth records for Mary and Patrick. I was not able to find birth or Baptism records for Bridget. In U.S. census records, Bridget and Mary are listed as sisters. Bridget's death record has the same father and the same maiden surname for her mother, but a different first name. This could be an error; I have found many errors on birth certificates. Or it could be that she was not a biological sister, but maybe a cousin, raised with the family. Just one theory. It remains a mystery. All four siblings emigrated to New York.
I believe there was at least one other sibling, James, who remained in Ireland. All evidence points to that, same father, townland, children who emigrated to U.S. indicate the above-mentioned Bridget as an aunt. But I haven't found birth or Baptism for James, and so I don't know his mother's maiden name which would tell us for sure.
Roger, thank you for assisting us last year!