Place turnips with water to cover in a medium saucepan; bring to a boil. Cover and cook 10 minutes. Add sugar; cook 5 minutes or until tender. Drain.
Melt butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until thickened. Stir in salt, pepper, and turnips. Transfer to a warm serving dish; sprinkle with parsley and serve.