In a Dutch oven or stockpot, cook bacon over medium heat until browned and crisp, stirring occasionally, about 8 minutes.
Add onion and carrots; cook until vegetables are soft, about 5 minutes. Add garlic and rosemary; cook until fragrant, 1 minute. Season with salt and pepper.
Stir in tomato paste, beans, broth, and 4 cups water. Bring to a boil, reduce heat, and simmer, partially covered, until soup has thickened slightly, about 10 minutes.
With a potato masher, mash some of the beans (still in pot) to thicken soup. Add pasta; cook until al dente, about 10 minutes.
To serve, ladle soup into bowls; sprinkle with parsley.
Use a vegetable peeler to shave a few Parmesan curls over each serving.