Melt the butter over low heat in a skillet. Whisk the eggs together loosely with a fork, then pour into the pan.
Allow the eggs to sit for 1-2 minutes before beginning to stir. Add the salt, then, with a wooden spoon, stir frequently (nearly constantly), until the eggs form curds, about 5-10 minutes depending on your stovetop.
Add the crème fraîche and parsley, and stir until the eggs are just solid.
Remove from the heat and pour into a warmed bowl for serving. The eggs will continue to cook slightly after being removed from the heat, so bear this in mind when deciding to remove them.