Place the eggs in a saucepan and cover with cold water. Bring to a boil over medium-high heat. Once boiling, cover and remove from heat. Let sit for 12 minutes. Transfer to an ice bath and let cool completely.
Peel the eggs and slice in half lengthwise. Pop the yolks into a bowl and set the whites aside.
Divide the egg whites between two bowls of dyed water, red food coloring in one and blue in the other. Let soak for 10 to 15 minutes until the desired color is reached. Remove and pat completely dry.
Mash the egg yolks until smooth. Add the mayonnaise, Dijon mustard, relish, salt, and pepper. Mix until creamy. Taste and adjust seasoning as needed.
Transfer the filling to a piping bag or zip-lock bag with a corner snipped. Pipe generously into each colored egg white half.
Garnish with fresh parsley and a pinch of red pepper flakes over each egg. Arrange alternating pink and blue eggs on a serving plate.