Heat oil in a large skillet over medium heat. Cook and stir beef and onion in the hot oil until browned, about 5 minutes. Stir in spaghetti sauce, zucchini, mushrooms, and carrot; season with salt and pepper. Simmer until flavors combine, about 5 minutes.
Spray half of each sheet of phyllo dough with cooking spray and fold in half lengthwise. Place 1/2 cup of the beef mixture on one edge of each phyllo sheet; top with 1 teaspoon Parmesan cheese. Fold up into rectangular parcels.
Transfer parcels to a baking sheet. Spray with cooking spray.
Bake in the preheated oven until golden brown, about 15 minutes.