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Boil red potatoes for 10 minutes. Drain and set aside.
Saute onions, celery, carrots in butter for a few minutes then add parsley.
Stir in flour, salt, and pepper.
Slowly add milk, stirring constantly. Heat until bubbly, about 12 to 15 minutes.
Toss in potatoes.
Finely chopped green onions work in lieu of parsley.
Red potatoes hold their form the best and do not get mushy as they cook.