There was no civil registration in Ireland before 1864 so there wouldn't be any civil birth record for Mary.
If she emigrated at 14, she wouldn't have been married in Ireland because she wasn't of legal age.
There are no census records in Ireland for the 1800s.
So you'd need to know specifically where Mary was born in County Cork, her parents names and religious denomination to find what church register for her specific birth location might have a baptismal record for her.
Church registers aren't centralised or computerised and their availability depends on the specific geographic location as well as the religious denomination.
There are nearly 6,000 towns and townlands in County Cork.
So her and her children's civil and church records in the US would be the source to find her parents names and specific birth location in County Cork.
Her civil and church marriage and death records, obituary, tombstone, cemetery record, will and possibly her US naturalisation records if she was naturalised would be sources to find the detail information you need.
Her children's civil and church birth, baptism, marriage and death records may include where their mother was born in County Cork.