In a large saucepan, bring potatoes, carrots, celery, onion, vegetable broth and salt to a boil. Reduce heat; cover and simmer until potatoes are just tender. Do not rinse: mash mixture slightly. Stir in milk.
In a small mixing bowl, blend butter, flour, parsley, and pepper; stir into potato mixture. Cook and stir over medium heat until thickened and bubbly.
Remove from heat: add cheese and stir until cheese is almost melted. Let soup stand for 5 minutes.