Salem is dog friendly and dogs are allowed in all parks, provided they are on a leash. It goes without saying that your dog needs to be properly trained before you should take them to a park. While Salem itself doesn’t have a leash law, for now, dogs are required to be on a leash while in a park.
Salem Oregon proper has two designated off leash dog area. There is one at Minto Brown Island park and another one at Orchard Heights Park. There is some talk about having an area at Cascades Gateway Park and North gate Park. There is an off leash dog area at Keizer Rapids Park, which is just north of Salem.
While not located in Salem, Silverfalls State Park has an excellent off leash area for dogs as well.