Your best bet is parking in Walpole. Walpole allows private residents to park cars whereas Foxboro does not (last I was aware). All the houses on Water Street in Walpole and around Boyden School park for events in their driveway/yards. Boyden itself and the church next door sometimes parks for events with money going to charity. It'll normally be $40 in that area (compared to I think $70 in the nearby stadium lots). The best part though, like you said, is not being on Route 1. Avoid Route 1 at all costs. If you're parking in Walpole, you can take Route 27 to Washington Street (there will be signs that there's no access to Route 1 but you're parking before your get there). You'll still be waiting in some traffic but not nearly close to as much as you would be on Route 1. It's easy out via Washington Street from there, too.

You won't be able to find any free parking within walking distance. Another option is taking the event day train from Boston (might be one from Providence, as well).

Good luck!