I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse and al most non-existant prior to 1800. Many early B/M/D events weren't recorded, or the records didn't survive.
For a Thomas Byrne and Bri(dget) Kelly M 1908 +/- 10 yrs, somewhere in Ireland, with the parents' names unknown, I found 0 records. Civil registration was mandatory after 1863. Most of these records survived and SHOULD be in the IFHF's database. Civil B/M/D records are kept by the General Register Office. Contact it at WWW.GROIRELAND.IE (fees apply).