Pat meat dry with paper towels; line a large platter with a double thickness of paper towels.
Pour enough oil into a large deep-sided pot to coat the bottom (4-5 tablespoons).
Set the pot over med-high heat.
When the oil is hot but not smoking, add enough pieces of meat to fit comfortably in a single layer (do not crowd).
Brown the meat on all sides, turning often, for 4-5 minutes.
Using a slotted spoon, transfer meat to paper towels to drain.
Repeat with the remaining meat, adding more oil if needed.
Remove pot from heat and wipe out any excess oil with a paper towel.
Return meat to pan and add beef stock, Worcestershire, soy sauce, 1 teaspoon salt, and several grinds of pepper.
Bring to a simmer over med-high heat, then decrease heat and cover; cook at a simmer for 1 1/2 hours.
Meanwhile, heat 3 tablespoons oil in a large skillet over med-high heat.
When hot, add the carrots, celery, parsnips, and onion; sauté until just slightly softened, for about 5 minutes.
Remove skillet from heat and set aside.
Heat 2 tablespoons oil in a medium skillet set over med-high heat; when hot, add the mushrooms and sauté, turning and adding more oil as needed, until lightly browned, 4-5 minutes; set aside.
After the meat has cook 1 1/2 hours, add the sautéed vegetables (but not the mushrooms), cubed potato, barley, and thyme to the pot; bring to a simmer, cover, and cook until barley and meat are tender, 45-60 minutes.
Stir occasionally, adding more water if consistency becomes too thick.
Add the mushrooms to the soup and simmer until heated through, 2-3 minutes.
Taste soup and season with salt and pepper, if needed.
To serve, ladle soup into soup bowls and garnish with chopped parsley.