In a medium skillet over medium heat, melt butter; cook potatoes and onion 10 to 15 minutes, or until tender. Add milk and pepper and heat until hot. Do not boil. Remove from heat and set aside.
In a medium bowl, combine flour, baking powder, salt, and water; mix well to form a stiff dough (you may need to use your hands). Roll dough into a 1/2-inch rope, then cut into 1/2-inch pieces.
In a soup pot over medium-high heat, bring chicken broth to a boil. Add carrot and drop dough pieces into broth. Cover, reduce heat to low, and cook 15 to 20 minutes, or until dumplings are light and fluffy in center. Add potato mixture and parsley to soup pot and heat 5 minutes, or until heated through.