I searched the ROOTSIRELAND.IE database. It contains abt 20M B/M/D. It doesn't cover all of Ireland yet. Co. Kerry, parts of Co. Cork, and Dublin City aren't in it. The Sullivan surname is highly concentrated in the Irish southwest.
For Thomas Sullivan B 1854 (exact) somewhere in Ireland, with the parent's names unknown, I found 14 records: 3 in Co. Cork (as expected), 1-Down, 2-Louth, 2-Waterford, 3-Limerick, and 3-Tipperary. You will need some other bits of info to determine which (if any) is the correct one.
If you know the forename of the Thomas' 1st born son, this MAY give you Thomas' father's name.