In a bowl, combine red onion rings and vinegar. Cover and refrigerate for at least 30 minutes.
While onions are pickling, make an aioli: in a bowl, combine mayonnaise, garlic cloves, lemon juice, chopped parsley, table salt and ground black pepper; mix well and set aside.
Grill burgers for 6 minutes per side, or to an internal temperature of 165ºF. Remove from grill.
Assemble each sandwich: Spread aioli on both sides of sliced buns. Place 1 burger on the bottom of each bun, followed by lettuce, pickled red onion, avocado, flake salt and cracked pepper. Top with other half of bun and serve