Equipment: an electric coffee/spice grinder or a mortar and pestle.
Bring water, vinegar, beet, shallot, sugar, bay leaf, and 1/2 teaspoon salt to a boil in a 2-quart saucepan, then simmer, covered, until beet is tender, about 20 minutes. Cool completely, uncovered. Put beet mixture in a container with eggs and marinate, chilled, gently stirring once or twice, at least 2 hours.
Finely grind caraway seeds in grinder.
Remove eggs from beet mixture and pat dry (discard beet mixture). Cut in half lengthwise and remove yolks. Mash yolks with mayonnaise, mustard, parsley, and half of caraway. Season with salt and pepper, then divide among egg whites. Sprinkle with remaining caraway.
Cooks’ notes: •Eggs can be marinated, chilled, in an airtight container up to 3 days.
•Eggs can be filled 2 hours ahead and chilled, loosely covered.