Break or cut the cracklings into bite-size pieces.
Beat the eggs in a bowl with salt and a generous grinding of black pepper.
Heat an omelet pan over medium heat. Add the butter and tilt the pan so that the butter coats the entire bottom and side of the pan. When the pan is warm and the butter is just beginning to sizzle, pour in the eggs. When the bottom is just set, lift the edge with a spatula or fork to allow the unset eggs to flow underneath.
When most of the eggs are set, put the duck cracklings in the center and slide and roll the omelet to the edge of the pan. Invert the omelet onto a warm plate. Garnish with the parsley.
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