I searched the Irish Family History Foundation's online database at ROOTSIRELAND.IE. It has abt 20 M B/M/D, civil, census, etc records for most Irish counties. Records before 1863 are increasingly scarce - and almost non-existant before 1800.
For Sarah Mullen B 1874 (exact), Co. Galway, with a father forenamed Luke and mother Sally, I found 1 record. A view (usually at a small fee) of the detailed record MAY reveal other details.
For a Luke Mullen and Sally Connelly marriage, Co. Galway, 1864 +/- 10 yrs, I found 0 records. This suggests the event wasn't recorded, hasn't been recorded, or the record hasn't survived.