The children’s births are all before the start of statutory birth registration in Ireland (1864) so you won’t find a birth certificate for John Parker Moore. You might find his baptism. You say he was possibly Church of Ireland. (That would seem to be confirmed by the fact that his sister Sarah married in St Johns Church of Ireland. Tradition was to marry in the bride’s church). If so, the bad news is that the Newtownhamilton Church of Ireland records pre 1871 were destroyed in the 1922 fire in Dublin, during the civil war. Baptisms for 1823 to 1826 only have survived and there’s a copy in PRONI (the public record office) in Belfast.
If the family were Presbyterian, then there are numerous Presbyterian churches in the area. Clarkesbridge Presbyterian has the following records: Baptisms, 1822-50; marriages, 1840-45; lists of communicants, 1822-62. Again there is a copy of these records in PRONI.
Newtownhamilton 1st Presbyterian church has baptisms and marriages from 1833 onwards but the only copy is still held by the Minister, so you would need to write to him to have them checked. PRONI do hold the following information from that church:
Register book containing list of session members, 1833, details of families by townland, 1833, accounts, 1832-36, stipend accounts with names, 1832-35, deaths, 1833-c.1836, and lists of poor people supported, c.1833.
Newtownhamilton 2nd Presbyterian church has the following records: Baptisms, 1823-1915; marriages, 1823-44 and 1862- 1936; new communicants, 1823-61; communion roll, c.1870-1938; session minutes, 1827-9, with censures, 1824-9; lists of membership certificates received and given, 1823-41; census of the congregation, 1826. There is a copy in PRONI.
There was also a Presbyterian church in Tullyvallen where your family appears to have lived. They don’t appear to have any records for the 1800s, but you would need to double check that with the church.
Griffiths Valuation suggests there was also seceder Presbyterian congregation in the townland too. I have no knowledge of any records for that. The Presbyterian Historical Society in Belfast might know if you want to contact them.
Griffiths lists a William Moore farming in Tullyvallen in 1864. A possible relation to John & Agnes/Nancy perhaps? William had plot 39 which was a 22 acre farm. There are also 3 Gordon households, headed by Anne, William & James.
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=nameSearch
Probate abstract: The Will of William Moore late of Tullyvallen County Armagh Farmer deceased who died 28 September 1880 at same place was proved at Armagh by John Carrol of Tullyvallen aforesaid Farmer one of the Executors.
The above will is on-line and can be read on the PRONI site. He left most of his estate to nephews and nieces:
http://apps.proni.gov.uk/DCAL_PRONI_WillsCalendar_IE/WillsSearch.aspx
According to the GRONI indexes he was 70 when he died.