Place the shrimp in a food processor and pulse until roughly chopped but not completely pureed.
In a mixing bowl, combine the chopped shrimp, breadcrumbs, mayonnaise, egg, minced garlic, parsley, Old Bay seasoning, salt, and pepper. Mix until well combined.
Form the mixture into four equal patties, about ¾ inch thick.
Preheat a skillet or grill over medium heat. Add a bit of oil if using a skillet.
Cook the patties for about 4-5 minutes on each side, or until they are golden brown and cooked through (internal temperature should reach 145°F).
Toast the burger buns on the grill or in a toaster if desired.
Assemble the burgers by placing a shrimp patty on each bun and adding your favorite toppings like lettuce and tomato.