Start by finely chopping the salmon fillets. In a mixing bowl, combine the chopped salmon, breadcrumbs, mayonnaise, Dijon mustard, lemon juice, garlic powder, salt, pepper, and parsley if using. Mix until well combined.
Divide the mixture into 4 equal portions and shape each portion into a patty, about 1 inch thick.
Heat a non-stick skillet over medium heat and add a little oil to coat the bottom. Once hot, carefully place the salmon patties in the skillet. Cook for about 4-5 minutes on each side, or until golden brown and cooked through.
While the patties are cooking, prepare the garlic aioli by mixing 1/2 cup mayonnaise with 1 clove minced garlic and a squeeze of lemon juice in a small bowl. Stir until smooth and well combined.
Toast the hamburger buns in a separate skillet or toaster until lightly golden.
Assemble the burgers by spreading garlic aioli on the bottom half of each bun, adding a salmon patty, and topping with lettuce, tomato, and onion. Place the top half of the bun on each burger and serve immediately.