Your post didn't contain any DOB, DOM, etc. This makes research tough and inconclusive.
I research the Irish Family History Foundation's database. It contains 12,000,000 surviving rcds for abt 65% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 35%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are ONLY available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome).
For Cat(herine) Sullivan M ????, Athea parish, Co. Limerick, I found 8 rcds: 1832, 1860, 1865, 1868, 1881, 1891, and 2-1894.
For Denis Cunningham M ????, ditto, I found 0 rcds. This infers that the 8 rcds for Catherine are likely all false positives. The marriage may have occurred elsewhere. Will leave it to you to search other Irish counties and/or America.
For Denis Cunningham B ???? ditto, with the father's forename unknown, I found 1 rcd in 1879. I'm not sure if this fits your unstated timeline. Father's forename was Cornelius.
For Cat(herine) Sullivan B ????, Athea parish, Co. Limerick, with the father's forename Pat(rick), I found 4 rcds: 2-1860, 1862, 2-1885. Note this excludes all of the civil B registrations after 1863.
For Pat(rick) Sullivan M ????, Athea parish, Co. Limerick, I found 4 rcds: 1830, 1841, and 2-1859.
For Mary Mullane M ditto, I found 1 rcd in 1859. This matches with a 1859 rcd for Patrick. These are likely for your ancestors. A view of the detailed rcds SHOULD show they married each other. If so, the rcds MAY reveal their parents' names and other info.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.