Heat the olive oil in a large nonstick skillet over medium heat. Add the onion, bell pepper, 1 teaspoon salt, and ½ teaspoon pepper. Cook, stirring occasionally, until very tender, 12 to 14 minutes.
Add the eggs and cook, stirring frequently, until the eggs are just set, 1 to 2 minutes. Add the Cheddar and cook, stirring, until just melted, 1 to 2 minutes more.
Remove from heat and stir in the tomatoes and parsley.
Drizzle the bread with oil and serve alongside the eggs.