My great grandparents (Vacca and Perardi) immigrated to Braidwood in the 1880-90s from the area in the Italian Piedmont known as the Canavese--an area northwest of Torino. I believe that almost all of the Italians who immigrated to Braidwood at that time came from this area called the Canavese. I also believe it is safe to assume that the Perardi's to whom you refer are related by blood, marriage, or both, to families Rossi, Doglio, Marchetti, Pomatto, et al.
If you would like to contact me I can email to you a transcription of Perardi baptisims that occurred in the Catholic church in Braidwood around the turn of the 20th century. Although I cannot answer your question, perhaps the names of the children, parents, and godparents will provide additional clues to aid in your search.