I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN parish, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Bri(dget) Carey B 1965 (exact) in Co. Mayo with the father forenamed Owen, I found 0 B rcds.
B/M/D registration was civil after 1863. Civil rcds, if they survived the Four Courts Fire of 1922, are kept by the General Register Office. Contact it at http://www.groireland.iehttp://www.groireland.ie for certificate copies. Fees apply.
For Owen Carey M 1860 +/- 4 yrs in Co. Mayo, I found 2 M rcds: 1859 and 1863. The 1863 rcd is the more likely. A view of the M rcds should list the name of the bride. This may validate the rcd if it exists.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.