I search the Irish Family History Foundation's database. It contains 11,500,000 surviving rcds for abt 60% of Irish counties. ALL Irish rcds B4 1840 are sparse.
For Mary Finan B 1885, I found 1 B rcd. A view of the B rcd SHOULD show her parents' names.
For Tom (Thomas) Finan M 1880 +/- 4 yrs, I found 6 M rcds. One of these MAY be your ancestor. If the parents' names are the same as shown on Mary's B rcd, then this rcd is likely that for your ancestors.
Suggest you google the Foundation's name, register, opt to be notified as counties are added, and browse (free).