Other options for finding names of parents of the original immigrants:
Circa 1910 passenger manifests for the U.S. listed not only the person the immigrant was bound for, but also closest relative left behind. In both cases the person listed may be a parent.
Various U.S. vital records for the original immigrants may list parents names.
Original Social Security Applications (SS5) listed parents names. Ancestry has a database of transcriptions of some of these.
