Had a look at the ancestry census transcriptions. Father Thomas born Ireland. Mother presumably deceased between 1848 and 1851, too early for statutory records.
Also had a look for births on family search to no avail.
Best bet would be to get marriage cert of young Thomas to from scotlandspeople.gov.uk
This will give parents' names of both the bride and groom.
There are 4 marriages for Thomas H*ney between 1863 and 1891 in Ayrshire. Search will cost 1 credit and image of cert 5 credits. 30 credits costs £7. Might be an idea to look at originals of the censuses.