Cook pasta according to package directions. Meanwhile, heat a large, wide pot over medium heat. Cook sausage, onion, and garlic, stirring frequently, until sausage starts to brown, about 5 minutes. Add chard and tomatoes and cook, covered, until chard wilts, about 4 minutes. Stir in cream.
Put drained pasta in a large, shallow serving bowl. Pour sausage and chard mixture over pasta and sprinkle with basil and parmesan cheese.