Remove and discard sausage casings.
In a 4- to 5-quart nonstick pan over high heat, combine sausages, onion, and garlic. With a spoon, break meat into small pieces and stir often until mixture is browned, about 15 minutes.
Meanwhile, break the lasagna into 2- to 3-inch pieces.
To pan, add broth, 2 cups water, tomatoes (with juice), tomato sauce, basil, oregano, and lasagna. Boil, stirring often, until pasta is tender to bite, about 15 minutes.
Ladle into wide bowls and sprinkle with cheese and parsley.
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