Combine water, potatoes, carrots, and salt in a large saucepan; bring to a boil. Reduce heat; cover and simmer 10 minutes. Stir in peas, cauliflower, and broccoli; cover and simmer an additional 10 minutes or until vegetables are tender.
Melt butter in a small Dutch oven over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in pepper. Gradually add vegetables and cooking liquid to white sauce, stirring well. Continue to cook, uncovered, over medium heat until thoroughly heated.
Ladle soup into individual bowls; sprinkle each with parsley. Serve warm.