Place the eggs in a pot and fill with cold water until the eggs are just covered. Bring the pot to a boil, then cover, remove from heat, and let sit for about 12 minutes.
Transfer the eggs to a bowl of ice water for about 3 minutes, then peel them and cut them in half.
Transfer the egg yolks to a bowl, and set the cooked egg whites aside.
Mix the mayonnaise, mustard, relish, salt, and pepper with the yolks and transfer to a piping bag. (Alternatively, use a zip-top bag with a corner cut off.)
Pipe the mixture into the egg whites, garnish with paprika and parsley, and serve chilled.
