I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
For Moore Wilson, I can't find any B or M rcds for this forename-surname combination anywhere at any date in Ireland.
Searching for the children B in Armagh, I found: John B 1821 +/- 2 yrs, 3 rcds: 1818, 1819, and 1822. Ann B 1825 ditto, 1 B rcd: 1825. M. (E.) B1828/29, 0 B rcds. Sarah B 1830 +/- 2 yrs, 1 B rcd 1830. M. B 1831 ditto, 0 B rcds. James B 1834 ditto, ditto. William B 1837 ditto, 2 B rcds.
For Sarah Conland B 1826 +/- 2 yrs, I found 0 B rcds.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse.
Research posted by others suggest that the LDS's Family History Library contains rcds which aren't in the IFHF database. Suggest you try this. Note, the National Archives of Ireland also has microfilm copies of parish registers.