I searched the rootsireland database. It contains 19 million+ records and is almost complete for all Irish counties.
For Patrick Conroy B 1831 +/- 1 yr, Co. Mayo, with parents' names unknown, I found 1 B record. This date is VERY early for Irish records. Many weren't created - few survived. This records MAY be for this ancestor. A view of the detailed record (at a small fee) MAY reveal his parents' names and other info.
For a marriage of Patrick Conroy and Mary Begley somewhere in Ireland at an unknown date, I found 1 M record in 1870, Co. Laois.
For a marriage of James Fallon and Rose Donnelly ditto, I found no M records anywhere in Ireland at any date.
For Ed(ward) or Ned Conroy marriage to Rose Fallon somewhere in Ireland at any date, anywhere, I found 1 record, 1900, Co Mayo. The groom's father's name was Patrick, but the bride's mother's forename either wasn't recorded, or isn't Rose (she may have died before the marriage).
These slightly positive results may be erroneous. Wish you had more definitive info as to dates or places.
Note - the rootsireland database isn't complete yet. Suggest you register with the IFHF at rootsireland.ie. It will notify you as records are added to it.