1. Bring the water, vinegar, beet, shallot, sugar, pickling spice, and 1/2 teaspoon salt to a boil in a small saucepan, then lower the liquid to a simmer, covered, until the beet is tender, about 20 minutes. Cool the liquid completely, uncovered. Put the beet mixture in a container with the eggs and marinate, chilled, gently stirring once or twice, at least 2 hours and up to 2 days. (The longer the time, the stronger the color and flavor will be, but also firmer the texture of the egg. So it’s up to you.)  

2. Remove the eggs from the beet mixture and pat them dry. Cut each egg in half lengthwise and remove the yolks. Mash the yolks with the mayonnaise, mustard, hot sauce, and half the parsley. Season the filling with salt and pepper, then divide it among the egg whites. Top with the rest of the parsley.