Chop salmon into 1/8- to 1/4-inch pieces. Transfer to a bowl with 2/3 cup parsley, 1/4 cup green onions, 2 tablespoons lemon juice, bread crumbs, seafood seasoning, 1/2 teaspoon salt, and 1/2 teaspoon pepper; mix together with your hands. Divide into 8 equal portions and pat into 3 1/2-inch patties (1/2 inch thick). Chill, covered, at least 30 minutes or up to 4 hours.
Stir together yogurt, mayonnaise, 2 tablespoons parsley, 2 tablespoons green onion, 1 tablespoon lemon juice, 1/8 teaspoon salt, and 1/8 teaspoon pepper in a small bowl.
Preheat an outdoor grill to medium heat (325 degrees F to 375 degrees F).
Brush patties with oil. Oil grate. Grill patties, turning carefully, until cooked through and a thermometer inserted in center of each patty registers 160 degrees F, 2 to 3 minutes per side. Serve on buns with sauce and arugula.