Add ⅓ of the salmon to the bowl of a food processor. Pulse until the salmon is smooth, about 10 to 15 seconds.
Add the remaining salmon, parsley, garlic, onion powder (if using), Dijon, mayonnaise, lemon juice, salt, and pepper into the food processor with the smooth salmon. Pulse 6 or 7 times, until everything is combined and there are still chunks of salmon visible.
Transfer to a bowl and gently fold in the roasted peppers and bread crumbs.
Divide the mixture into 4 mounds, and then form each into a patty, about 3 to 4 inches in diameter. A hamburger press or large cookie cutter is helpful here, but not totally necessary.
Chill the patties for at least 20 minutes, or up to 24 hours.
If grilling, turn your grill to medium-high heat. Lightly spray a grill pan, (or non-stick skillet) with cooking spray. Use a spatula to transfer the patties onto the grill pan and place over direct heat on the grill. Cook for 7 minutes and then carefully flip and cook for another 7 minutes. If cooking on the skillet, follow the same timing. The internal temperature of the burgers should be 135°F to 145°F.
Meanwhile, butter the buns and toast during the final cooking of the salmon burgers.
Serve at once on toasted buns, with lettuce, tomato, onion, and aioli, if desired.