In a food processor (or by hand), finely chop the parsley, garlic, and oregano. Reserve 1/4 cup of the parsley mixture to use another day and transfer the remainder to a bowl. Add the oil, vinegar, salt, and pepper to the bowl and mix well.
Preheat the broiler. Cut the skirt steak into 4 portions and place in a roasting pan or on a baking sheet in a single layer. Rub both sides of the steak with the parsley sauce. Broil for 3 to 4 minutes per side for medium, remove, and let rest, covered, for 5 minutes before serving. Serve the remaining steak with the raw onion rings.