Heat a large Dutch oven between medium and medium-high heat. Add 1 tablespoon olive oil; once the oil is shimmering, add half the chicken sausage and cook until lightly browned on the first side, about 2 minutes. Use tongs to flip the sausage pieces and cook until lightly browned on the second side, about 2 minutes. Use tongs or a slotted spoon to remove the browned sausage to a bowl. Repeat with the remaining olive oil and chicken sausage, but do not remove the second batch of sausage from the Dutch oven.
Add back the first batch of sausage along with the chicken broth, Parmesan rind, cannellini beans, lemon juice, a large pinch of salt and a couple grinds of black pepper. Increase the heat to high and bring to a boil, then use a wooden spoon or spatula to scrape up any browned bits at the bottom. Reduce to a strong simmer and cook for 5 minutes to allow the flavors to come together. Stir in the kale and cook until just wilted, about 2 minutes. Remove and discard the Parmesan rind. Season the soup with additional salt and pepper if needed.
Ladle the soup into bowls, then sprinkle with parsley and Parmesan. Serve with lemon wedges and crusty bread.