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