Bring a large pot of lightly salted water to a boil. Add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes.

Meanwhile, heat oil in an extra-large skillet over medium-high heat. Add onion; cook until softened, 2 to 3 minutes. Add asparagus, mushrooms, and squash; cook until just tender, about 5 minutes. Add tomatoes, carrot, garlic, oregano, black pepper, salt, and red pepper flakes; cook until tomatoes begin to soften, about 1 minute.

Drain penne; stir into vegetable mixture along with 1/4 cup Parmesan cheese. Top servings with remaining cheese and serve with lemon wedges.