Heat grill to medium-high. In a small bowl, combine the parsley, oil, oregano, vinegar, red pepper, and ¼ teaspoon salt. Set aside.
In a food processor, pulse the salmon 3 or 4 times, just until coarsely chopped (still somewhat chunky).
Add the scallions, ½ teaspoon salt, and ¼ teaspoon pepper and pulse to combine. Form the mixture into ¾-inch-thick patties.
Oil the grill grate. Grill the patties, turning once (do not press or flatten), until opaque throughout, 2 to 4 minutes per side.
Place the burgers on the buns and top with the chimichurri sauce.