Heat the oil in a Dutch oven over medium-high heat. Add the onion and cook, stirring often, until soft, 5 to 6 minutes. Add the tomatoes and their juices and cook, stirring occasionally, for 3 minutes.
Stir in the squash, thyme, 5 cups water, 1 1/2 teaspoons salt, and 1/4 teaspoon pepper and bring to a boil. Reduce heat and simmer until the squash is tender, about 15 minutes. Stir in the beans and spinach and cook until the spinach is wilted and the beans are heated through, 2 to 3 minutes.
Meanwhile, prepare the biscuits according to the package directions, sprinkling with the Parmesan before baking. Serve with the soup.
Tip: To make this satisfying soup even heartier, add 1/2 cup orzo or other small pasta at the same time you add the water.