Heat oven to 400° F. Heat the oil in a large ovenproof nonstick skillet over medium-high heat. Add the onion and cook, stirring occasionally, for 3 minutes. Add the bell peppers and 1/2 teaspoon salt and cook, stirring occasionally, until the vegetables are soft, 8 to 10 minutes.
In a medium bowl, whisk the eggs, 1 teaspoon salt, and 1/4 teaspoon pepper. Mix in the parsley. Pour over the vegetables in the skillet, stir once, and crumble the cheese over the top.
Transfer to oven and bake until the center is set, 18 to 20 minutes. Slide out of the skillet and cut into squares before serving warm or at room temperature.
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