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).
For Sarah Burns B 1874 (exact), Co. Antrim & Down, with a father forenamed John, I found 1 church rcd. However, the parish isn't Newry. Will leave it to you to find this. A view of the detailed B rcd SHOULD show her parent's names as well as other details.
For John Burns M 1844 +/- 4 yrs, Co. Antrim & Down, I found 20 rcds - too many to summarize here. None were in 1842.
For Bri(dget) McShane M ditto, I found 1 rcd in 1842. This is a NO match. It is likely that the M event's rcds didn't survive.
Observation - your estimated M date seems too early to still be having children roughly 30 yrs later. You may want to verify your info, or try other search ranges.
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.