The Roman Catholic church records for Coalisland (Armagh) diocese begin from 1861 (baptismal records) and 1862 (marriage records). These can be accessed at the National Library of Ireland, Pos. 5583, or The Public Records of Northern Ireland (PRONI) at MIC.1D/34, or the Church of Latter Day Saints ref. no. 0979709 item 3. If you have any difficulty, you could try writing to the parish priest.
Civil registration records start from 1864 and can be found at the General Register Office (GRO) here: http://www.groireland.ie/research.htm
Baptism records should contain the names of the parents, the date, the name of the child and the name of sponsors. Most registers also records the residence of the parents.
You could also try checking the land records called the Tithe Applotment Books (1823-38) or the later Griffth's Valuation (1848-64). Griffith's is freely available here: www.askaboutwww.failteromhat.com Failte Romhat has lots of other useful links you could try looking at!