When cool enough to handle, tap the eggs lightly on a hard surface to crack the shells all over, then remove the shells. Make sure you get every piece of shell off of the eggs. Cut the eggs in half lengthwise. Carefully remove the yolks from the whites and place them in a small bowl; set the whites aside.

With a fork, mash the yolks together with the mayonnaise, mustard, and salt. Stir in the parsley (or sprinkle the parsley on top after you fill the whites). Using a small spoon, scoop the yolk mixture into the whites.

Serve at room temperature.