I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties. Records before 1863 are increasingly sparse: almost non-existant before 1800. Protestant records are scarce at any date.
For Mary Carney B ???, somewhere in Ireland with a father forenamed John, and mother Mary, I found 137 records. However, none has a mother surnamed Duffy.
For Bridget Mulcahey, B ???, Co. Limerick, with the parents' names unknown, I found 160 records. Without either the parents names or an exact DOB, it will be difficult to verify which, if any, is the right one. Suggest you search censuses and/or obituaries to find more info.