Whether dogs are permitted off-leash, permitted only on leash, or prohibited in a particular site is shown in the list of site features on this website.
On-leash Dogs 
Dogs on leashes are allowed in all parks and conservation areas in Newton, except inside tot lots and Little League fields.
No dogs are permitted in the Newton Cemetery & Arboretum.
Off-leash Dog Areas 

Off-leash dog areas exist in the following locations:
- the “old”, or southern, end of Cold Spring Park
- Hunnewell Park (but not in the adjacent conservation area)
- five other city parks and playgrounds
To use one of the off-leash areas, a dog must be licensed and must wear an off-leash medallion, as described on this page of the City of Newton website. Information about each of the seven off-leash parks can be found on the city page.
The creation of the off-leash dog areas was discussed in this 2010 article.
Dogs in Conservation Areas
Rules for dogs in Newton conservation areas are on this page.
Dogs in State Reservations
Rules for dogs in Massachusetts state reservations, including Hammond Pond Reservation, Cutler Park, and the Blue Heron Trail along the Charles River are here.