Hi Elizabeth,
Aw, thanks--I was only saying that I would love to help you out as I am a bit nosey at the same time!

Where did her parents get married in? Since you have their marriage entry it should tell you where.
If that is true, then the database we were looking at are not your Mary Carney.
I think she may be born in Newry which is in Co Down (I must hammer the map of NI in my head as I often got the places mixed up, lol).
This site only covered Co Antrim and Co Down along with Belfast.
Kind regards,
Tees