Preheat the oven to 400˚. Brush a 24-cup mini-muffin pan (preferably nonstick) with olive oil. Toss the potatoes, tomatoes, eggplant, olive oil, garlic, thyme, 1 teaspoon salt and a few grinds of pepper in a large bowl.
Place a potato slice in each muffin cup (trim if needed). Top with an eggplant slice, then a tomato slice. Cover the pan with foil and bake until the vegetables are tender, 30 to 35 minutes.
Uncover the pan, sprinkle with the parmesan and bake, uncovered, until browned, about 15 more minutes. Let cool 5 minutes, then remove from the pan. Top with chopped parsley.