It will be very difficult to find their parents.
First I assume you've looked at every Census in the U.S. they were in. Sometimes parents move back in with their children.
I don't know anything about Ireland birth records but you probably don't have enough information to know if you find your James Duffin.
I'd look for their marriage record especially if they married in the USA. If you find the record and the parents aren't on it, I'd look for the newspaper announcement of their marriage.
I'd contact the Willson-Wicklund REsearch Center & Archives, www.olmstedhistory.com/resources/research-center.html
who has Rochester newspapers 1859-1912. They also might have additional information about the family. Successful Searching!