Place 1/4 of vegetables in center of a sheet of foil. Drizzle with butter; top with basil. Sprinkle with salt and pepper.
Bring up sides of foil over vegetables; fold down two times. Double fold both ends of foil to seal packet, leaving room for heat circulation inside. Repeat to make four packets. Place packets on baking sheet with 1-inch sides.
Bake 15 to 18 minutes on the cookie sheet.
Open packets carefully by cutting beside top fold with a sharp knife, allowing steam to escape. Sprinkle 1 tablespoon Parmesan cheese over vegetables in each packet.