Your research will have to start in the US, as that is where you'll find clues. An index for their marriage:
Name: Martin Walsh Birth Date: 1860 Birthplace: U. S. Age: 24 Spouse's Name: Mary Quigley Spouse's Birth Date: 1860 Spouse's Birthplace: Spouse's Age: 24 Event Date: 02 Nov 1884 Event Place: Paterson, Passaic, New Jersey
Getting a copy of the original might name the parents. Is this him in the 1910 US Census:
Paterson, New Jersey
Walsh, Martin F., aged 49, married, 26 years, b. Ireland, parents b. same, immigrated 1865, boilermaker, locomotive shop Walsh, Mary A., aged 49, married, 26 years, 3 children born, 2 children living, b. New York, parents b. Ireland Walsh, Michael, aged 23, single, b. New Jersey, boilermaker, locomotive shop Walsh, James, aged 22, single, b. NJ, laborer, jute mill
1900 Census has them in Paterson, indexed as Welsh. Michael's birthplace is given as Pennsylvania which suggests that they didn't settle in Paterson right away.