Rinse eggplant, cut each crosswise into 3/4-inch rounds, and discard stems and smooth ends. Lay slices side by side on a large tray or 2 or 3 baking sheets, about 12 by 15 inches.
In a small bowl, mix oil and Italian herb mix. Brush cut sides of eggplant with oil mixture.
Cut mozzarella cheese into thin slices, making 1 piece for each eggplant round.
In a 1 1/2- to 2-quart pan over medium heat, stir marinara sauce occasionally until warm.
Set eggplant rounds on a grill over a solid bed of medium-hot coals or medium-high heat on a gas barbecue (you can hold your hand at grill level for only 3 to 4 seconds). Close lid of gas barbecue. Cook until eggplant is very soft when pressed, creamy inside (cut to test), and browned, 6 to 8 minutes; turn once. Set marinara sauce on cool area of grill to keep warm.
Top each eggplant round with a slice of mozzarella cheese. Cover barbecue until cheese softens, about 1 minute. Transfer to a platter. Spoon marinara sauce onto slices, then sprinkle with parmesan cheese and parsley. Season to taste with salt and pepper.