Statutory registration of non RC marriages started in Ireland in 1845. I don’t see a marriage for Joseph & Eliza in the statutory records 1845 – 1860 so I would guess they married before that. In which case you need to search church records, which are probably not on-line.
Tradition was to marry in the bride’s church (after which she’d attend her husband’s). The Greenaways around Tandragee were nearly all Church of Ireland (in the 1901 census) and so I’d look in the local COI parish records. That appears to be Ballymore. Their baptism & marriage records go back to 1783. There’s a copy in PRONI, the public record office in Belfast.
Birth registration didn't start till 1864 so Thomas & Joseph junior won't have birth certificates. You may find their baptisms but again probably not on-line. The Whittens around Tandragee seem to have been a mix of Church of Ireland and Presbyterians. There are 5 Presbyterian churches in the area. Again copies of their baptism records are in PRONI.
Joseph Whitten married Jane Martin in 1878 in Partick, Glasgow. You can view that marriage certificate on-line on the Scotlandspeople site, for a small fee. The certificate should tell you whether Joseph senior & Eliza were dead or alive at that date. If dead it will say “dcd” after their name(s). If alive there will be just the name. (Scottish Ministers and Registrars were pretty meticulous about that). It should also record Joseph’s occupation which may help too. If you let me have all that information, I’ll search for deaths in Ireland whichever side of 1878 is appropriate.