Preheat the oven to 350 degrees F. Lightly grease a deep 9x9-inch baking dish or similar 3 1/2-quart casserole dish with nonstick spray.
Salt the eggplant first, which helps mellow its flavor. Before starting the recipe, cut the eggplant into 1/4-inch thick rounds. Spread the slices on paper towels, then lightly sprinkle them with salt. Let sit 10 minutes. Pat the slices dry with additional paper towels, then proceed with the recipe as directed.
Quarter the zucchini then cut into 1/2-inch slices and place in a large mixing bowl (each piece of zucchini should be roughly 1/2 to 3/4 inches in size). Next, slice the eggplant into 1/4-inch rounds, then stack the rounds and cut into roughly 3/4-inch pieces. Add to the bowl with the zucchini. Halve the cherry tomatoes and add them to the bowl. Drizzle the cut vegetables with the olive oil, then add the garlic, salt, pepper, 1/3 cup of the Parmesan cheese, and half of the basil and parsley. Toss gently to combine.
Transfer the vegetables to the prepared baking dish. Bake for 30 minutes, cover the pan with aluminum foil, then continue baking for 30 additional minutes, until the vegetables are tender. Sprinkle with the remaining Parmesan cheese, basil, and parsley. Serve warm.