Peel the eggplant, cut off the stem and cut lengthwise into 3"x1"x3/8" slices. Trim off zucchini ends. Cut into slices about the same size as the eggplant. Place vegetable slices in a large non-aluminum bowl (glass or plastic are fine). Toss with 1 teaspoon salt; let stand 30 minutes. Drain and pat slices dry on paper towels.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Cook eggplant and zucchini slices in batches until lightly browned, about 1 minute per side, adding more olive oil as needed. Set vegetables aside.
Cook onions and bell peppers in the same skillet in 2 to 3 tablespoons oil until tender but not browned, about 10 minutes. Stir in garlic and season with salt and pepper to taste.
Peel tomatoes while onions and pepper cook by dipping in boiling water, then ice water to loosen the skins. (Or use a serrated-edge peeler.) Cut out tomato stems, cut tomatoes in half and squeeze out the seeds and excess juice. Slice tomato pulp into 3/8-inch strips. Lay tomato strips over the onion and peppers in the skillet. Season with salt and pepper.
Cover skillet; cook over medium-low heat until tomatoes begin to render juice, about 5 minutes. Place 1/3 of the tomato/onion mixture in a Dutch oven or heavy casserole. Sprinkle with 1 tablespoon parsley. Arrange half of the eggplant and zucchini on top. Top with half of the remaining tomato mixture and parsley. Top with the remaining eggplant and zucchini slices. Finish with the remaining tomato mixture and parsley.
Cover and simmer over low heat for 10 minutes. Uncover, tip casserole and baste with rendered juices. Adjust seasonings, if necessary. Raise heat slightly. Cook, uncovered, until juices have evaporated, about 15 minutes. Stir often to keep vegetables from scorching on the bottom.