I am trying to trace my ancestors that came to the USA from County Tyrone, Ireland in 1836...a Mrs. P. Cullen that came with her two sons: James, age 7(born March 10, 1829) and Michael, age 4. A third son, Peter, was born at sea (July 4, 1836)on the voyage to USA. Mrs. P. Cullen was also accompanied by a Thomas Cullen, age 30 (relationship unknown - not spouse) and possibly a daughter, Mary.
The births are all long before the start of statutory birth registration in Ireland (1864), and so to trace the family you would need to rely on church records. To do that we’d really need some idea of where in Tyrone the family lived and what denomination they were.
You might be able to get parent s names from marriage or death certificates. That would help ensure that if you do trace them, you have the right family. Cullen is a common name. Looking at the 1901 Irish census there were 227 of them in Tyrone alone. (Nearly all were RC).
My Cullen family was RC, also. Other than that, in all of the research I have done I have never found the first name of the Mrs. P. Cullen (my great-great-grandmother) that came to the USA with her sons and what happened to her after she arrived here. I only have record of her sons and the Thomas Cullen that came with them after they made their way to Wisconsin, USA (We think about 1848 - the first actual record was James' marriage certificate issued in 1863 at Milwaukee, WI). None of the documents I have seen (marriage, death certificates, etc.) state the parents' names or anything more specific than they came from Tyrone, Ireland. I recently purchased a membership to Ancestry; but as you stated, Cullen is a very common name and my research using their records has not produced any results.