In a large saucepan, cook kielbasa over medium heat until lightly browned, about 5 minutes; drain and set aside. In the same pan, combine the potatoes, corn, broth, celery, carrot and seasonings. Bring to a boil. , Reduce heat; cover and simmer until vegetables are tender, about 15 minutes. Add the milk, cheese, parsley and sausage. Cook and stir over low heat until cheese is melted and soup is heated through, about 5 minutes.