Mix the ground turkey, mayonnaise, Worcestershire sauce, onion powder, garlic powder, parsley, salt, and pepper together in a large bowl. Form the mixture into 4 patties, about ½ inch thick. Make a small indentation in the center of each patty so that the burgers cook evenly.
Preheat air fryer to 360°F and spray the basket with oil.
Place the burger patties in the air fryer in a single layer. Depending on the size of your air fryer, you may need to cook the burgers in two batches. Cook 5 minutes, then flip the burgers over and cook another 3-5 minutes until cooked through. The final cooking time will depend on your air fryer so the best way to tell if the burgers are cooked is to check them with a meat thermometer. The temperature should be 165°F.
Before removing the burgers from the air fryer, top each one with a slice of cheese. Close the air fryer and let the cheese melt, about 30 seconds. Remove burgers.
To make the Burger Sauce, mix the ketchup, mayonnaise and pickle juice together in a small bowl.
Place turkey burgers on warmed hamburger buns. Top with desired toppings like tomato, avocado and/or onion slices. Serve burgers with Burger Sauce.