Melt butter in a large pot over medium heat. Sauté onions and celery in hot butter until tender, 5 to 10 minutes.

Stir in water and potatoes. Bring to a boil and simmer until potatoes are tender. Add cheese and stir until melted.

Stir milk and flour together in a medium bowl until flour is mostly dissolved; pour slowly into soup, stirring constantly. Cook until soup is thickened, about 5 minutes.

Portion soup into bowls and garnish with parsley to serve.