Preheat the oven to 350 degrees F.
Heat the olive oil in a large nonstick skillet set over medium heat. Add the onions and carrots. Season with ¼ teaspoon salt. Cook, stirring occasionally, until softened, about 5 minutes.
Stir in the garlic, Italian seasoning and red pepper flakes. Cook for 1 minute.
Add the crushed tomatoes, chickpeas and remaining ¼ teaspoon salt, bring to a boil, then simmer for 10 minutes.
Remove from the heat and stir in the cooked quinoa and 3 tablespoons parsley.
Place the pepper, cut side up, in a large baking dish. Divide the chickpea mixture evenly between the peppers. Pour ½ water in the bottom of the baking dish.
Cover tightly with foil and cook until the peppers are tender, about 30 minutes.
Sprinkle the mozzarella cheese and Parmesan cheese over each pepper. Bake, uncovered, for an additional 5 minutes.
Garnish with remaining 1 tablespoon of parsley. Serve.