Heat oven to 375°F. In 12-inch skillet, heat oil over medium-high heat until hot; cook onions in oil 15 to 20 minutes, stirring frequently, until onions are soft and golden brown. If onions start to stick, stir in 1/4 cup water.
Meanwhile, in small bowl, mix mayonnaise and horseradish. Slice sheet of dinner rolls in half horizontally; place bottom half in ungreased 13x9-inch (3-quart) baking dish. Spread cut side of roll top with horseradish mayonnaise. Mound roast beef on each roll bottom. 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 19 to 22 minutes or until rolls are golden brown and cheese is melted. Cut into individual sliders, and serve.