Email the National Archives Branch in New York... they should be responsible for the naturalization records or will tell you where to search.
You should pursue this and get a copy of the original Declaration and the Petition.... and any other documents associated with the file. Tell them or send them the letter you have, and the application should answer lots of questions about this family, including information on how they entered the US.
The fee for them copying the original papers should be minimal, and you might also get to see a photograph attached to the papers as well. Both parents might have these applications..... so ask for both.