Spread the onions over the bottom of a 5 1/2- to 6-quart slow cooker and top with the carrots, garlic, white beans, thyme bundle and sausage links. Mix the diced tomatoes with the broth and 3 cups water and pour over the sausages. Add the Parmesan rind if using.
Cook on high for 4 to 5 hours or on low for 7 to 8 hours; the beans will be tender and begin to fall apart.
Uncover the slow cooker, remove and discard the thyme bundle and Parmesan rind and transfer the sausage links to a cutting board. Stir the pasta into the stew and continue to cook, covered, until the pasta is cooked through, about 20 minutes.
Turn off the heat. Cut the sausages into bite-size pieces and stir into the stew along with the parsley and vinegar. Season with salt and pepper. Serve with grated Parmesan on the side for sprinkling on top and crusty bread for soaking up the broth.
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