Preheat oven to 375º and line an 5"-x-8" loaf pan with parchment paper. In a large skillet over medium heat, heat oil. Add onion, celery, carrot, and mushrooms and cook, stirring occasionally, until vegetables are soft and most of the liquid has cooked out, 6 to 8 minutes.
Using a potato masher in a large bowl or food processor, mash the chickpeas until a rough paste forms. (A few large pieces of chickpea are OK.) Transfer to a large bowl if using a food processor.
Add cooked vegetables, bread crumbs, parsley, soy sauce, and Worcestershire sauce to the bowl with chickpeas. In a medium bowl, whisk together ketchup and barbecue sauce. Add half of this mixture to the bowl with the chickpeas. Season with paprika, salt, and pepper, and stir until all ingredients are evenly incorporated.
Transfer chickpea mixture to prepared loaf pan, packing the mixture in gently. Smooth top, then brush with half of the remaining ketchup mixture and bake for 30 minutes. Remove from oven, brush with remaining ketchup mixture, and bake 30 minutes more.
Let cool 10 minutes, then garnish with parsley and serve