Catmong,
There’s no simple way to confirm Joseph Mawhinney’s parentage. But I would start by looking at Joseph’s townland at the time of his marriage, and his occupation. Also his father’s occupation. Then what information do you have about Thomas? Where was he living when he died? Was it the same townland that Joseph lived in? Did Thomas leave a will, and if so did it mention son Joseph? You have to weigh up what you find. I assume this is Joseph in 1901:
http://www.census.nationalarchives.ie/pages/1901/Down/Magheralin/Ballymackeownan/1242688/
That indicates he was a labourer. They are notoriously difficult to research because they can often move about a bit.
Is there a family gravestone? Does it mention father and son?
If the family were Church of Ireland and attended Magheralin church, then the following records exist, in PRONI in Belfast:
“Baptisms, 1692-1871; marriages, 1692-1782 and 1785- 1845; burials, 1692-1863; vestry minutes, 1692-1902; annotated transcript of vestry minutes, 1702-1855.
Extracts from diocesan records relating to Magheralin parish, 1600-1850.
Extracts from parish registers, 1784-1853.
Subscription list relating to the building of the church, c.1840; pew register, 1844”.
Children to Joseph & Elizabeth, all registered in Lurgan:
Elizabeth 4.7.1879
Thomas 12.8.1881
Jane 20.8.1883
Margaret 1.1.1886
Robert 19.9.1887
George 26.9.1892
David 15.9.1891
Joseph 27.1.1896.
So there was a child named Thomas, who was the first born son. If the family followed the traditional naming pattern, then that fits with grandfather Thomas. Looking at Jane’s birth in 1883 in Drumo & Drumcro, it’s obvious they did move about a bit. That won’t make it easy. They only moved to Ballmakeownan in 1901 according to the Valuation Revision records.
Likewise with the search for John Baxter, what occupation and townland appears on the 1855 marriage cert? What was the father’s name? Have you found them in Griffiths? Did he leave a will etc? You say there is a baptism where the mother’s name looks like Cecelia? Where did you find that? Have you searched for other children to that same couple, to clarify the mother’s name? Have you searched for a death for her?