Heat a charcoal or gas grill or the broiler to moderately high heat and put the grill rack about 4 inches from the heat source or the broiler rack 4 to 6 inches from the heat source.
Peel the eggplant if the skin is thick. Cut into 1/2-inch- thick slices. If you like, stir the garlic into the oil. Brush one side of the eggplant slices with the oil. Place, oiled side down, directly on the grill or on a baking sheet. Sprinkle with salt and pepper, then brush with more oil. If you’re broiling, put the baking sheet under the broiler.
Grill or broil, turning once, until the eggplant has browned and become tender, 5 to 10 minutes per side, brushing with more oil if it looks dry. Transfer the slices to a platter, drizzle with the remaining oil, sprinkle with parsley, and serve hot, warm, or at room temperature.