Preheat the oven's broiler and set the oven rack in the upper third of the oven. Beat the eggs and egg whites in a mixing bowl until smooth. Whisk in the parsley, 1/4 teaspoon salt, and 1/4 teaspoon pepper until evenly blended; set aside.
Heat the olive oil in a 10-inch cast-iron skillet over medium heat. Stir in the onion, and season with 1/4 teaspoon salt, and 1/4 teaspoon pepper. Cook and stir until the onion softens and turns translucent, about 5 minutes. Stir in the tomatoes, and cook an additional minute. Add the cooked Swiss chard; cook and stir until the Swiss chard is hot. Pour in the egg mixture, cover, and cook until the edges of the egg have set, about 5 minutes.
Uncover the frittata, and sprinkle with Parmesan cheese. Broil in the preheated oven until the cheese is golden brown and the center of the frittata has set, about 2 minutes. Remove from the oven, and let stand 3 minutes before serving.
