Melt butter in large saucepan over medium heat. Add onion and cook until tender.
Add carrots and celery and cook for 10 minutes or until carrots become a little tender. Add water or broth, potatoes, salt, and pepper. Bring to a boil and allow to simmer for 15 to 20 minutes or until all vegetables are tender.
Add milk and parsley and heat just to boiling.
In a separate bowl stir cornstarch into cold water. When completely dissolved add to hot soup, stir and serve.