I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
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 Mic(hael) Mulloy M 1862 +/- 1 yr, Co. Mayo, I found 2 rcds: 1861, Bekan RC parish, and 1862, Kilconduff-Swinford RC parish.
For Jud(ith) Corrigan (+variants) M ditto, I found 0 rcds within a meaningful range. My failure to find a correlated M rcd to those for Michael casts doubt on the findings for him. They may have married outside Mayo, or their rcd didn't survive.
For Mic(hael) Mulloy B 1843 +/- 4 yrs, Co. Mayo, with the father's forename unknown, I found 6 rcds: 1839 Burriscarra RC parish, 1843 Addergoole, 1844 Aglish, 1845 Addergoole, 1845 Kilconduff-Swinford, and 1847 Bekan. Father's forenames were Thomas, John, Patrick, Patrick, Patrick, and Michael respectively. The 1843 rcd seems to fit your criteria best. However, there are no B rcds in Addergoole for Brigid and Mary.
For Jud(ith) Corrigan (+variants) B 1845 ditto, I found 1 rcd in 1841, Islandeady RC parish. Father's forename was Patrick.
Suggest you get the names of their offspring and search for their B rcds. These have a higher likelihood of surviving.
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.