Set the oven to preheat at 375°F. In a 10-inch nonstick sauté pan, warm 2 tablespoons of olive oil over medium-high heat. Add the minced garlic, chopped onion, and shredded carrot, cooking for approximately 5 minutes until softened. Introduce the crushed tomatoes along with basil and parsley, stirring to combine. Allow the mixture to simmer for 15 minutes until it thickens slightly, then season with salt and black pepper to taste. Transfer the tomato mixture into a 2-quart baking dish, spreading evenly across the bottom. Arrange the eggplant, zucchini, and Roma tomato slices upright in a circular pattern over the sauce. Lightly brush the vegetables with the remaining olive oil. Cover the dish with foil and bake for 30 minutes. Remove the cover and continue baking until the vegetables are tender and the top is slightly browned. Serve promptly.