1. To make the aioli, mix 3 tablespoons marinade with the mayonnaise in a small bowl until smooth; set aside.
2. Combine salmon, egg, parsley, scallions, breadcrumbs, lemon zest and juice, 1/2 teaspoon salt, and 1/2 teaspoon pepper in the bowl of a food processor. Pulse until mixture is very finely chopped and comes together, about 30 pulses. Using wet hands, gently form mixture into four patties of equal size, each about 3/4" thick; transfer to a plate and set aside.
3. Heat oil in a large cast-iron or nonstick skillet over medium heat until shimmering. Add salmon patties and cook until bottoms are golden brown, about 4 minutes.
4. Carefully flip and continue cooking until second side is golden brown and patties are cooked through, about 4 more minutes; transfer to a clean plate.
5. Toss arugula with remaining 1 tablespoon marinade and season with salt and pepper to taste. Spread aioli on bottom half of each bun and top with salmon patties and arugula. Place remaining bun halves on top and serve.