Stir in mixed vegetables and mushrooms; then add day-old rice, ketchup, soy sauce, and 1/4 teaspoon salt. Mix well until heated through.
In a mixing bowl, whisk together 4 eggs with 2 tablespoons milk and 1/4 teaspoon salt. Pour into another non-stick skillet over medium-low heat. Cook gently until edges set but center remains slightly runny.
Place a portion of fried rice on one half of the omelet; fold the other half over to create a dome shape. Slide onto a plate.
Drizzle with ketchup and sprinkle with minced parsley before serving.