Combine all ingredients in a medium size saucepan. Heat, stirring, until butter melts and sauce is well blended. Season with salt and pepper. Cool. Reheat before using.
Arrange corn, eggplant slices and onions on a large baking sheet. Arrange peppers, zucchini, asparagus and carrots on another baking sheet. Separate ¾ of sauce and reserve it in a bowl. Brush vegetables, on both sides, with remaining sauce.
Grill corn, eggplant and onion at medium heat until tender. Turn and brush occasionally with the sauce. Transfer vegetables to a platter as they are cooked. Cover with aluminum foil to keep warm. Grill peppers, zucchini, asparagus and carrot, basting frequently until tender. Transfer to the same platter. Season with salt and pepper.
Decorate with parsley sprigs.
Heat reserved sauce and serve with vegetables.