Heat oven to 400°F. Place tortillas on 2 large baking sheets. Cook, flipping once, until crisp, about 10 minutes; set aside.
Meanwhile, heat oil in large skillet over medium heat. Add mushrooms, peppers, onions and garlic. Cook until vegetables are soft and tender, about 10 minutes; set aside.
Spread tortilla crust with 2 tbsp. tomato sauce, ¼ cup vegetable mixture, 2 tbsp. mozzarella cheese and ½ tsp. parmesan cheese; repeat with remaining crusts and toppings ingredients.
Transfer pizzas to same baking sheets. Cook until cheese is melted, and edges of tortillas are golden brown, about 10 minutes.