In 12-inch skillet, melt butter over medium-low heat. Stir in brown sugar. Add onions; cook 12 to 16 minutes, stirring occasionally, until onions begin to turn golden brown. Place onions in medium bowl; cover to keep warm.
In same skillet, cook beef patties over medium heat 6 to 8 minutes, turning once, until meat thermometer inserted in center of patties reads 160°F.
Return onions to skillet; pour gravy over onions and patties. Cook 2 to 3 minutes longer, stirring once or twice, until gravy is thoroughly heated. If desired, sprinkle with chopped fresh parsley.