I have to admit I am getting confused with all the double names but...
I have a copy of the History of Placer and Nevada Counties in which my great grandfather, Luigi Pardini has a short bio. The bio includes the info that his father, Giovanni Pardini remarried to Louise Boccia and one of the second set of children is a Joseph. Joseph had 2 sisters, Adele and Angelina, and a brother Benedicto.
As for dates of this all I can pin down from the bio is that Luigi's first wife, Clementina Giorgio died in 1869 or early 1870. So the second marriage would have been in the 1870's. I would be happy to send you a copy of this page.
Unfortunately, all of my great-aunts and uncles are gone, but I have asked a few of my cousins to rack their brains.
I will also go back to my Ellis Island file, because Joseph sounds familiar.
Let me know if this helps.