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 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 Mary Heaney B ???? in Co. Longford with a father forenamed James, I found 6 rcds: 1834, 1838, 1855, 1863, 1882, and 1884. Take your pick.
For Mary Whitney M ???? in Co. Longford, I found 11 rcds: 1835, 1839, 1840, 1849, 1853, 1859, 1860, 1861, 1883 and 1885.
For James Heaney M ???? ditto, I found 11 rcds: 1816, 1829, 1833, 1834, 1843, 2-1862, 1864, 1874, 1883, and 1887. The only one in common with Mary's is in 1883. This seems too late for your timeline.
For port of entry, Google "New York Passenger Lists 1820 - 1957." This leads you to several databases, some free, some fee. Also Google "NARA" for naturalization papers.
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.