I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 70% 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 Mary Canty B 1855 +/- 1 yr somewhere in Ireland with the father's forename unknown, I found 1 rcd 1855 Co. Cork, Cobh RC parish. Father's forename was Michael. This date doesn't correlate with the mother's immigration date of 1850-1852.
For Michael Canty M 1851 +/- 4 yrs, Co. Cork, I found 0 rcds. The closest date was 1843.
For Ellen Canty B 1826 +/- 1 yr somewhere in Ireland with the father's forename unknown, I found 1 rcd in 1825, Co. Cork, Coachford RC parish. Father's forename was Jeremiah.
For Jere(miah) Canty M 1822 +/- 4 yrs somewhere in Ireland, I found 0 rcds.
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.