I searched the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. B4 1840 rcds are sparse, B4 1800 rare. My search is limited to this database.
For Jane Freeborn B 1812 +/- 1 yr in Co. Armagh, I found 0 B rcds. Extending the search to all Ireland, I got the same result.
Suggest you look for her marriage rcd in the U.S. This MAY show the names of her parents. Passenger rcd keeping in 1836 wasn't great.
Google the Foundation's name to lead you to this database. Pls post your search results to my genforum msg.