In a large skillet with a fitted lid, heat the olive oil over medium heat. Add the peppers and onions and cook until the peppers are almost soft, about 8 minutes. Add the tomatoes and basil and cook until all the vegetables are soft, about 10 minutes. Add salt to taste.
With the back of a spoon, create 8 indentations in the vegetables. Carefully crack the eggs in the indentations. Don’t let the eggs touch the bare bottom of the pan or the sides or they will burn or stick. Reduce the heat to medium-low, cover the pan, and simmer until the whites of the eggs are set, 6 to 8 minutes. Season with salt and pepper and sprinkle with the Parmesan and parsley. Serve promptly.