Preheat the oven to 350 degrees F (175 degrees C). Lightly grease an 8-inch square baking dish.
Whisk flour, baking powder, and 1 teaspoon salt together in a bowl. Whisk eggs, milk, and butter together in a separate bowl. Pour egg mixture over flour mixture; beat until well blended. Stir in parsley; set aside.
Heat a large skillet over medium-high heat; stir in ground beef and onion. Cook and stir until beef is crumbly, evenly browned, and no longer pink, about 10 minutes. Drain and discard any excess grease.
Combine drained meat mixture with celery, carrot, 1/2 teaspoon salt, pepper, Worcestershire sauce, and steak sauce. Spread meat mixture evenly into the prepared baking dish. Pour Yorkshire pudding batter over meat.
Bake in the preheated oven until the pudding has risen tall and is golden, about 45 minutes. Serve immediately.