I searched 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 parish, 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 James Fitzgibbon B 1826 in Limerick, I found 1 B rcd. Will leave it to you to find his parish.
For Margaret Sullivan B 1831 +/- 4 yrs in Limerick, I found 42 B rcds.
For James Fitzgibbon M 1841 in Bulgaden, Limerick, I found 1 M rcd.
For Maria Sullivan M ditto, ditto.
For William Fitzgibbon B 1811 +/- 4 yrs in Limerick, I found 0 B rcds. "Source" data indicates some problems with the rcds for Fedamore.
For Ellen Fitsgerald B 1823 (exact) ditto, I found 2 B rcds.
For Kate Fitzgibbon B 1841 (exact) in Limerick, I found 1 B rcd in Bulgaden.
From the database, I identified the following children and their DOB as likely siblings: Pat'k 1844, John 1846, Joh. (possibly Johanna) 1848, Ellen 1850, Bridget 1855, and Maria 1858, James 1861 and Margaret 1864. View the detailed rcds to validate the parentage. When you enter the forenames in the search screen spell them EXACTLY as shown above.
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.