Sprinkle eggplants with salt and layer in a colander. Weigh down with a plate and set aside to drain for 20 minutes. Brush salt off eggplant with paper towel and pat dry.

Preheat a gas or charcoal grill, or place a stove-top grill pan over medium-high heat. Lightly brush all vegetables with oil, then grill, turning once, until tender and slightly charred, about 30 seconds per side for tomatoes, 2 - 3 minutes per side for eggplant, squash, and zucchini.

Whisk together oil, vinegar, parsley, and basil in a mixing bowl. Season to taste with salt and pepper. To assemble stacks, place 1 eggplant slice on a plate, then top with a tomato slice, a second eggplant slice, a squash slice, and a zucchini slice. Repeat process to use up all vegetables, then drizzle vegetable stacks with dressing and top with diced red peppers.