Boil pasta in salted water until al dente; reserve 1 cup pasta water.
Sauté garlic and onions in olive oil until fragrant.
Add mushrooms, bell peppers, broccoli, and corn; cook until tender.
Melt butter in the same pan, whisk in flour to make roux.
Gradually add milk while whisking until sauce thickens.
Stir in cream and cheeses off heat until melted and smooth.
Return vegetables to sauce, add pasta, and toss well.
Adjust consistency with pasta water; season to taste.
Garnish with parsley and extra Parmesan before serving.