I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
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).
Note 4 - Microfilms of parish church rcds are kept by the National Library of Ireland (NLI).
For Elizabeth Millar B 1829 +/- 4 yrs, northern Ireland, with the father's forename unknown, I found 5 rcds: 1826 Antrim, 1828 Derry, 1829 Derry, 1831 Down, and 1832 Derry. Father's respective forenames were Alexander, William, Joseph, Hugh, and Thomas. One of these MAY show the father's occupation was "Crofter." The ancillary data as to father's occupation was rarely rcd'd, especially at this early date.
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.