Heat 1 1/2 tablespoons oil in a large pot over medium-low heat. Add the onions and garlic and cook until soft, about 5 minutes.
Meanwhile, in a large skillet, heat another 1 1/2 tablespoons oil. Add zucchini in batches and sauté until slightly browned on all sides. Transfer each batch to the pot with the onions and garlic.
Working in batches, sauté eggplant, yellow squash, and bell peppers, one batch at a time, adding oil with each batch and transferring them to the pot when sautéed.
Season the ratatouille with salt and pepper. Add thyme sprigs and bay leaf. Cover and cook over medium heat for 15 to 20 minutes.
Add tomatoes and parsley. Cook, stirring occasionally, for 10 to 15 more minutes. Remove bay leaf and adjust seasoning before serving.