Heat a large skillet over medium-high heat. Cook and stir sausage, breaking into smaller pieces, in the hot skillet until browned and crumbly, 5 to 7 minutes. Mix noodles into sausage.
Pour crushed tomatoes and tomato sauce over sausage-noodle mixture; bring to a boil. Cover skillet, reduce heat, and simmer until noodles are cooked and sauce is thickened, about 25 minutes. Add spinach, ricotta cheese, and mozzarella cheese; cover skillet and remove from heat. Let lasagna stand for 5 minutes; top with Parmesan cheese.