Adjust oven rack to upper-middle position and preheat oven to 450°F. Toss the tomatoes, eggplant, and zucchini with 2 tablespoons olive oil, 1 teaspoon thyme, garlic, ½ teaspoon salt, and ¼ teaspoon pepper.
Pat chicken breasts dry with paper towels and sprinkle with the remaining 2 teaspoons thyme, ½ teaspoon salt, and ½ teaspoon pepper. Brush the remaining 1 tablespoon oil over a hot sheet pan, place chicken skin-side down on one side, and spread vegetables on the other side. Roast until the chicken releases from the pan and vegetables begin to wilt, about 10 minutes.
Flip the chicken skin-side up, stir the vegetables, and place lemon quarters cut side down on the sheet. Roast, stirring vegetables occasionally, until the chicken registers 160°F and the vegetables are tender, about 10–15 minutes.
Remove the sheet from the oven, tent with aluminum foil, and let rest for 5 minutes. Transfer the chicken to a serving platter. Toss the vegetables with the pan juices, season with salt and pepper to taste, and transfer them to the platter. Sprinkle parsley over the vegetables and serve with roasted lemon wedges.