Melt the butter in a large saucepan over medium heat and saute the onion and celery until softened. Stir in the flour and cook for 3 minutes, stirring frequently. Gradually add the stock, whisking until you have a smooth liquid. Add the potatoes, corn, thyme, salt and pepper. Cover and bring to a boil. Then reduce the heat and simmer for 15 minutes or until the vegetables are tender, stirring occasionally.
Stir in the crabmeat and heavy cream and heat the soup gently but thoroughly. Do not let it boil or the soup may curdle. Sprinkle with the chopped parsley and serve immediately.