1. Preheat grill to high heat using both burners. After preheating, turn the left burner to medium heat (leave right burner on high heat).
2. Combine first 5 ingredients and 1/4 teaspoon salt, stirring with a whisk.
3. Combine oil and paprika in a small bowl. Brush oil mixture evenly over mushrooms, tomatoes, eggplant, and radicchio. Sprinkle evenly with remaining 1/4 teaspoon salt and pepper.
4. Arrange vegetables on right side of grill rack (over high heat); grill vegetables 8 minutes or until just tender, turning once. Move any vegetables to left side of grill rack (over medium heat) if they get done more quickly. Place vegetables on a platter. Sprinkle with chopped parsley, if desired. Serve with sauce.