I can't find a marriage certificate for his parents either. I assume they married, because my Thomas has his name.
Not a safe assumption
Thomas Sr. seems to have died between 1855 and 1861, because Margaret is unmarried in the 1861 census
If she had been married to Thomas Lindsay and he had died, she would normally be listed as widowed, not unmarried.
I have his own marriage certificate listing his father as Thomas Lindsay, Joiner, Deceased.
How,
exactly, is his mother's name recorded on his marriage certificate?
I see from the index at SP that Thomas Lindsay and Helen Kyle were married in 1880. Which rather scuppers my idea of looking for a Thomas Lindsay who was a carpenter or joiner or wright in the 1881 census
On the other hand, Thomas Jr could have bent the truth a little when he got married, or perhaps his mother had told him that his father was dead when in fact he wasn't .... there is a Thomas Lindsay, joiner, aged 50, born Barony, with wife Catherine and assorted children. They look like the family of Thomas Lindsay and Catherine Brown, who were married in Glasgow in 1856. Their eldest son was Andrew, and there is a family in Shettleston in 1851 with head Andrew Lindsay and son Thomas, 22, joiner journeyman, born Barony. Note that this is pure speculation on my part and not to be interpreted as definite confirmation that this particular Thomas Lindsay is the errant father of your Thomas Lindsay.
I see, incidentally, that in 1881 Thomas Jr gives his age as 27. If accurate this means a birth in 1853 or 1854 rather than in 1855. You would expect a schoolmaster to be able to get his own age correct!
Des Garrity's 1861 census index lists Agnes Clelland, 53; Margaret Hill, 22; Thomas Lindsay, 6; and Ann Clelland, 63, all born Cambusnethan except Ann who was born in Bothwell. If accurate, that gives a birth between April 1854 and March 1855 for Thomas; it also means that Margaret was very young indeed to be the mother of Thomas.
The 1851 census transcription at
https://www.freecen.org.uk/cgi/search.pl lists
Agnes Cleland, head, 43, unmarried, sewster, born Cambusnethan
Robert Cleland, brother, 44, unmarried, shoemaker, born Cambusnethan
Margaret Hill, 12, born Cambusnethan
plus a niece, nephew and great-nephew.
If Margaret's middle name was Young, this points to Agnes being the daughter of Thomas Clelland and Margaret Young, baptised in Cambusnethan on 15 April 1808.
The only source that might reveal anything more about Thomas Lindsay Sr is the minutes of either the Cambusnethan or the Shotts Kirk Session. If Margaret Young Hill was a member of the congregation and unmarried, and her pregnancy came to the notice of the Kirk Session, there could be a record of her being summoned to answer for her sin, and that would normally include information about her partner.
The Kirk Session minutes are in the National Records of Scotland. They have been digitised, but are not (yet) available online, so until they do come online (which has been 'soon' for some time!) they can only be consulted in the Historical Search Room in General Register House in Edinburgh (not in the Scotland's People Centre there) or in a small number of archives elsewhere including Glasgow, Hawick, Aberdeen and Inverness. So unless you can go to one of those in person, or get someone to go on your behalf (quite a big 'ask' for a volunteer as it could involve scrolling through a large number of pages of digital images of dodgy handwriting) you will need to wait until they do become available online.