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 1816 +/- 3 yrs, Co. Cork, with the father's forename unknown, I found 3 rcds: 1813 Macroom RC parish, 1818 Inniscarra RC, and 1818 Donoughmore RC. Due to the early date and the fact that many of Co. Cork's rcds haven't been transcribed, the collective chance that any of these rcds are for your ancestor is about 20%. You may want to wait until the database is more complete.
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.