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 Jane Kirk M 1873 +/- 4 yrs, Co. Antrim, I found 3 rcds: 1869, 1870, and 1876.
For ???? Millar M 1869/1870/1876, I found 88 rcds, roughly 1/2 of which are for males. You can further discard many false positives by finding only those in the same parishes as found for the three for Jane.
For Jane Kirk B 1852 (exact), Co. Antrim, with a father forenamed William, I found 0 rcds.
For William Kirk M 1848 +/- 4 yrs, Co. Antrim, I found 1 rcd: 1846.
For Mary Anderson M ditto, I found 2 possible 1846 rcds. Will leave it to you to search by parish to see if either is a match for William's.
For William Kirk B 1823 +/- 4 yrs, Co. Antrim, with the father's forename unknown, I found 4 rcds: 1822, 1823, 1824 and 1825.
For Mary Anderson B 1825 ditto, I found 10 rcds: 1824, 2-1825, 2-1826, 2-1827, 2-1828, and 1829.
The names sound Scots-Irish and hence protestant. These rcds are most notable for their general absence. Those that I found are POTENTIAL matches. These merit further screening at the parish level to see if there are matches in the same parish. If so, the one(s) are very likely to be for your ancestors (no guarantee).
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.