The RC parish for Mullaghmore Co. Sligo would have been Ahamlish.
Records for the parish are on NLI MicroFilm, but like most of these films only cover up to 1879/1880.
Unless you are dealing with a very common surname it may be better to search for civil records.
p.s. transcripts for this parish appear to be included on Rootsireland (pay-website), with baptism and marriage records up to about 1900.
S.