Can the parotid saliva duct suffer damage from multiple tooth extractions?

Unlikely: Stensen's duct travels from the parotid gland through the buccal fat pad & enters the oral cavity around the second molars - of the buccal mucosa & not around the extraction sites of the teeth. However, conceivably, if an unusually large, & overly lateralized incision was made and/or tissue reflected, the duct could have been transected, crushed, or had partial obstruction due to edema post-op.

Rare at best: I have never heard of this happening, although anything is possible. More likely a result of an infection or something else rather than as a complication of the extractions.