In a heavy pot, add 3 cups water, potatoes, onion, and celery; bring to a boil, reduce heat, and simmer until potatoes are cooked through.

Add the milk and heat on low until hot, but do not boil.

Add salt and pepper to taste and then add the hard-boiled eggs, butter, and parsley.

Serve as soon as the butter has melted, stirring to mix the butter and parsley before ladling into individual bowls.