I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly scarce and almost non-existant prior to 1800. Early B/M/D events either weren't recorded, or the records haven't survived.
For Mary Gunny B 1815 +/- 5 yrs, Co. Cork, with the parents' names unknown, I found 6 records: 2-1810, 1-1812, 1-1814, 1-1815, and 1-1816. ONE of these MAY be for your ancestor, or all false positives. Your chances at this early date are about 50:50. A view of the detailed records (usually for a small fee), SHOULD show the parents' names and other lesser details if they were found in the original records.
Mary MAY have followed the Irish Naming Tradition. If so, the parents' forenames MAY be echoed in the first names for her children born in Australia.
Hope these clues help. Good luck and happy new year.