Melt butter in a Dutch oven over medium-high heat. Cook chicken in butter 15 to 20 minutes or until brown on all sides. Drain butter from Dutch oven.
Return chicken to Dutch oven. Stir water, carrots, onion, and salt into chicken. Heat to boiling; reduce heat. Cover and simmer 40 to 50 minutes or until chicken is no longer pink when centers of thickest pieces are cut. Prepare parsley dumplings.
Remove chicken and keep warm. Mix flour, milk, and pepper until smooth; pour into Dutch oven. Heat to boiling, stirring constantly. Boil and stir 1 minute and then reduce heat. Return chicken to Dutch oven.
Drop dumplings by rounded tablespoonfuls onto the hot chicken. Do not drop directly into the liquid. Cover tightly and cook 12 to 14 minutes or until dumplings are fluffy and dry on top. Serve hot!
For the dumplings: Mix flour, parsley, baking powder and salt in a medium bowl. Cut in butter, using a pastry blender, until mixture looks like fine crumbs. Stir in milk until all flour is moist.