I think the eldest son Thomas born circa 1829 England (per 1841 census) was the Thomas baptised 19/6/1829 Gateshead, Durham which follows on from the marriage you've found in Durham 1827.
Now it gets a bit confusing for me - you said at the outset that Thomas Park moved around, born Scotland, 3 children born in England, another born in Scotland, and that he also went to Northern Ireland. I found a Thomas Park on the 1861 census - right sort of age, born Scotland, a widower, with 2 sons born in Ireland (later census' show this to have been in Buncrana, Co. Donegal, N.I.) - however, this Thomas is a Foreman at Gas Works. I 'assumed' this was still your man and his career as a customs officer had ended.
By 1861 a Thomas (now widowed) 54 born Scotland and sons all seem to be in Bristol. There's a Thomas Park who married a Mary Dunstan on 10/6/1858 Bristol, son of a Thomas Park, and feel sure that this is Thomas junior as the following census shows he was born in Gateshead. The fact that the Thomas born 1829 Gateshead is also in Bristol convinced me that the older Thomas was indeed his father, together with 2 sons born in Northern Ireland where we know your Thomas went. The 2 sons born in Ireland being William John ca.1845 and Charles Crawford bc.1847 - however, there are some trees on Ancestry that just show these 2 boys stating parents as Thomas Park and Helen McKay married 1837 in Lanarkshire. But there are no sources shown and the births/baptisms of the 2 boys clearly not found which begs the question of how they knew the mother was named Helen McKay?? Did they simply find the marriage of a Thomas Park to a Helen McKay and presume they were the parents of said William John and Charles Crawford?? Why no children between 1837 and 1844?? It just seems a little odd to me.
So now I'm confused by the assumption I have made in that the Thomas in Bristol (died 1866) is your man when perhaps he isn't. It just seems a huge coincidence that the son Thomas b.1829 Gateshead appears in Bristol as the same time the 1861 census one does.
Whilst I see that all 3 Park boys mentioned above all married in Bristol it's not possible to view the original records online. Only know fathers name for all 3 is Thomas.
It looks to me like your Thomas and Elizabeth had:
Thomas bp.19/6/1829 Gateshead, Durham
Mary ca.1831 - possibly the Mary Jane bp.10/4/1831 Kirkby Lonsdale, Westmorland
Elizabeth Ellison bp.22/3/1835 Lancaster, Lancashire
Charles Crawford Clarkson bp.20/11/1836 Liverpool (died?)
Ann bc.1839 Linlithgow, West Lothian
plus more in Northern Ireland, one being another Charles Crawford ca.1847??
Just seems too much of a coincidence that another Thomas born the same sort of time in Scotland had a son Charles Crawford in N.I. Surely they must be the same Thomas especially as Elizabeth Ellison Park also married in Bristol in 1857?
Annette