Heat oven to 375°F. In 12-inch skillet, over medium-high heat, heat oil; cook onions in oil 15 to 20 minutes, stirring frequently, until onions are soft and golden brown.
Separate sliced beef from gravy in packages. Roughly tear beef slices into smaller pieces. Reserve gravy.
Slice sheet of dinner rolls in half horizontally; place bottom half in ungreased 13x9-inch (3-quart) baking dish. Spread cut side of roll tops with horseradish sauce. Layer roast beef pieces on bottom layer of rolls. Spoon caramelized onions over beef; top with Swiss cheese. Place top half of rolls over cheese.
In small bowl, stir melted butter and parsley until well mixed. Brush over top of rolls. Bake 20 to 22 minutes or until rolls are golden brown and cheese is melted.
Warm reserved gravy. Cut into individual sliders and serve with warm gravy for dipping.