Heat oven to 375 degrees. Heat oil over medium and cook onion until tender and lightly golden, 6 to 8 minutes.
Arrange the onion on the bottom of a 9-by-13-inch baking dish. Overlap tomato, squash, and potato on top of the onion. Season with salt and pepper, sprinkle with thyme and Parmesan, and drizzle with more oil.
Bake covered for 30 minutes. Uncover and bake until golden, 30 minutes more.