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 Pat(rick) McAuliffe B 1870 (exact), Meelin parish, Co. Cork with a father forenamed Mich(ael), I found 1 rcd.
For Daniel McAuliffe B 1868 (exact), Newmarket parish, ditto, I found 1 rcd.
For Michael McAuliffe B ????, Newmarket parish, Co. Cork, with a father forenamed Daniel, I found 1 rcd in 1840 within a relevant range.
For Michael McAuliffe M 1864 +/- 4 yrs, Newmarket, Co. Cork, I found 1 rcd in 1867.
For Martha McConnell M ditto, I found ditto.
The above are highly likely to be for your ancestors; they match your info precisely. Will let you search the database for the other siblings.
For Dan(iel) McAuliffe M 1836 +/- 4 yrs, Co. Cork, I found 7 rcds: 1832, 1833, 1834, 1837, 1838, 1839 and 1840.
For Johanna Barry M ditto, I found 18 rcds - too many to list here. Visit the database and find which ones match with the years for Daniel. Then compare these at the parish level of detail to see if there is a match there.
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.