The town-owned Keystone Lot (at Eastern Ave and High St) has 48 additional public parking spaces in addition to the metered street parking. This has made visits to the Square much more convenient for residents and out-of-towners alike. 
On Friday and Saturday nights if the Keystone Lot is full try Town Hall parking lot on Bryant Street. You can park there Friday nights, all day Saturday, Saturday nights and Sundays.

There are also several parking spots in the Dedham Square Crossing lot (at the intersection of High and East Streets across from the new clock tower) where you can park anytime.