I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 20 M church, civil, census, etc records for MOST Irish counties. Records before 1863 are increasingly sparse - almost non-existant before 1800. Co. Monaghan rcds are in this database; however there are a few problems with info for this Co. - lots of events weren't recorded.
For a Eliza Cleary and Owen Cusick marriage, 1857 (exact), Co. Monaghan, with the parents' names unknown, I found just ONE record. This MAY be for your ancestor. You need to visit the database and view the record to see if the mm/dd/yyyy match your info. If so, then the record MAY show their parents names and sometimes other info.