Heat 1 tablespoon olive oil in a large, deep skillet or Dutch oven. Add the turkey sausage, onion, fennel, salt, and pepper. Cook, breaking apart the meat, until it is browned and the vegetables are soft, about 8 minutes. Add the tomato paste and garlic and cook until fragrant, about 1 additional minute.
Lightly coat the inside of a 6-quart or larger slow cooker with nonstick spray. Add the browned sausage mixture. Next, add the carrots, 6 cups chicken broth, diced tomatoes in their juices, and Italian seasoning. Stir to combine. Cover and cook on LOW heat for 3 hours, until the carrots are nearly tender.
Stir in the kale and tortellini. Cover and cook on LOW for 30 additional minutes, or until the kale is wilted and tender and the tortellini is al dente. If your slow cooker runs hot, check the slow cooker a little early to ensure the pasta does not overcook. Stir in the red wine vinegar. If you would like a thinner, brothier soup, stir in additional broth until your desired consistency is reached. Serve hot, sprinkled with Parmesan and parsley.