Preheat oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3-4 minutes until translucent.
Add minced garlic and cook for 1 minute until fragrant.
Add ground turkey to the skillet and cook, breaking it up with a spoon, until browned and cooked through, about 6-8 minutes.
Add diced bell pepper, zucchini, and mushrooms to the skillet. Cook for 5 minutes until vegetables are tender.
Stir in the drained diced tomatoes, dried oregano, dried basil, salt, and black pepper. Cook for an additional 2 minutes.
Remove skillet from heat and stir in the cooked brown rice until well combined.
Transfer the mixture to a greased 9×13-inch baking dish.
Sprinkle shredded mozzarella and cheddar cheeses evenly over the top.
Sprinkle grated Parmesan cheese over the cheeses.
Cover the baking dish with aluminum foil and refrigerate if making ahead, up to 24 hours.
When ready to bake, preheat oven to 375°F (190°C) if not already heated.
Bake covered for 25 minutes, then remove foil and bake for an additional 15 minutes until cheese is melted and bubbly.
Remove from oven and let stand for 5 minutes before serving.