Elizabeth McLean’s birth in 1867 contains the information that her parents married in Ballymena in June 1837. The marriage records are probably not on-line. If you wanted to track it down you would need to get someone to search the relevant records in PRONI in Belfast. I would guess the family were Presbyterian (incidentally pointing to Scottish origins a couple of hundred years previously). If so, there were about 3 Presbyterian churches in the town at that time. Worth checking the Church of Ireland as well.
Birth registration didn’t start till 1864 in Ireland so if looking for the children's baptisms, they may be in those same records.