I research the Irish Family History Foundation's database. It contains 12,700,000 surviving rcds for abt 72% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Co Donegal rcds were added to this database July 7, 2009.
Note 1 - Without a KNOWN county, 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 Rose McKinley B 1844 +/- 4 yrs, northern Ireland, with the father's forename unknown, I found 1 rcd in 1842, Co. Antrim. The father's forename was Henry. This doesn't quite match your specifications. The fore and sur names sound Scots-Irish. Rcds from northern Ireland are not that great, especially for protestant denominations.
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.