Crack twelve eggs, and separate the yolks and egg whites into separate bowls.
Preheat oven to 300°F (150°C).
Spray a mini muffin tin with nonstick spray. Pour each yolk and egg white into its own mini muffin tin hole.
Place the tin into a baking dish with a tinfoil ring, wire wrack, or a towel to keep the muffin tin from touching the bottom of the baking dish directly (so the eggs steam gently in the oven and don’t turn brown).
Cover the eggs and the baking dish tightly with aluminum foil. Make sure the foil tents up so it doesn’t touch any of the eggs.
Lift one edge of the foil and pour in boiling water ⅓ way up the muffin tin and re-cover with foil.
Place the eggs in the oven for 13 minutes. Check to make sure the egg whites and yolks are both fully cooked.
Remove foil and muffin tin from baking dish and let cool.
With a thin knife, remove the egg yolks and egg whites from the tin. Set the egg whites aside.
In a bowl, combine the cooked egg yolks, Greek yogurt, relish, mustard, hot sauce, salt, and pepper. Mix with a fork until well combined.
Place the mixture in a piping bag with a desirable tip and pipe the yolk mixture onto the cooked egg whites. Sprinkle with paprika and chopped parsley.