USING A 1 3/4-INCH COOKIE CUTTER, cut out rounds for the centers of the bread slices. Spread mayonnaise on each of the bread rounds with a small spatula.
The diameter of the onion slices should approximate that of the bread rounds. Sprinkle each with salt and pepper. Place 1 onion slice on each half of the rounds.
Cover each one with another bread round, mayonnaise side down, on top of the onion. Using the spatula, spread the outside rim of the sandwich with mayonnaise, coating fairly liberally.
Roll the rim of each sandwich in parsley to coat.