Spray 3 1/2 to 4-quart slow cooker with nonstick cooking spray. Combine zucchini, mushrooms, eggplant, tomatoes, sugar and basil in slow cooker.
Cover; cook on high setting for 4 hours or on low setting for 8 hours.
Stir in oil and parsley. To serve, ladle mixture into soup bowls. Sprinkle with cheese.