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 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 Martin Slattery B 1867 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 8 rcds: 1863 SLI, 2-1864 GAL, 1864 TIP, 2-1866 GAL, 1867 MAY, and 1869 TIP.
For Mary Slattery B 1885 (exact) somewhere in Ireland with a father named Martin, I found 0 rcds.
For Thomas Slattery B 1887 ditto, I found ditto.
For John Slattery B 1888 ditto, ditto.
It appears that the children weren't born in Ireland, or their rcds haven't been transcribed yet.
For John Leehive B 1833 +/- 4 yrs somewhere in Ireland with the father's forename unknown, I found 0 rcds. In fact, I couldn't find ANY B or M rcds for this surname at anydate, anywhere in Ireland. This suggest an English origin.
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.