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The Fethard RC records go back to 1806, however, the records are not on the subscription site Roots Ireland. You would have to go to Ancestry.com or FindMyPast to look for Fethard record. I did look at FindMyPast and did not see a baptismal record for Richard Byrne or Jane Dunphy. I did find a Tithe Defaulter list record for a Richard Byrne https://www.findmypast.ie/transcript?id=IRE%2FENEC007%2F25163 Hopefully you can see the page. It appears that a Richard Byrne in Fethard could not pay his Tithe tax. This Richard had nine children and could not be the Richard who married Jane Dunphy but possibly it was his father or uncle.
Here is an excerpt:
Number of occupied houses110
AffidavitSent to them, would not pay. The County being so disturbed did not think it safe to go to harsh measures which even if done would have proved useless- for no person would purchase anything seized. Your Memorialist begs to state that having a family of nine children and his means but small, and for the last year and a half being deprived of his income, of course, he is heavily in debt which circumstance he hopes will be kindly taken into consultation in his aid of the Relief to be now afforded.