Cut eggplant horizontally into 1/2 inch thick slices. Place slices on baking sheet and sprinkle both sides with 3 teaspoons salt. Allow to stand for 30 minutes; rinse and drain well. Brush eggplant slices with 4 tablespoons oil and place on grill over low heat. Cook until lightly browned on both sides.
Cut eggplant into julienne pieces about 2-inches by 1/2 inch and place in large bowl. Add vinegar, remaining salt, red onion, parsley and pepper; toss well. Cover salad and refirgerate for about 3 hours. Toss again before serving on lettuce leaves.
Though served chilled, this recipe begins by cooking eggplant on the grill.