Your post looks familiar. I may have already answered it. If so, please excuse the redundancy.
I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
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 available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome), and the National Library of Ireland on microfilm.
For John Carroll B 1829 +/- 1 yr, St. John's RC parish, Co. Limerick, with a father forenamed Andrew, I found 1 rcd, 1829. The rcd may have info which sheds light on the family.
For Mic(hael) Carroll B 1834 ditto, I found 1 rcd, 1834.
For An(drew) Carroll M 1819 +/- 10 yrs, Co. Limerick, I found 1 rcd, 1811, Pallasgreen & Templebredon RC parish.
For Jane Keane M ditto, I found NO matching rcd. It looks like the rcds didn't survive, haven't been transcribed, or they married elsewhere.
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.