Cut the beef into 3/4-inch cubes, discarding any tough pieces of fat or gristle. Place in a bowl and season with 1 teaspoon salt and ½ teaspoon black pepper. Stir to coat. Let stand at room temperature while you chop the vegetables (this will help the beef soften).
Chop the vegetables: dice the onion and celery. Peel the carrots and parsnips and cut into ½-inch chunks. Mince the garlic.
In a Dutch oven or similar large, sturdy pot, heat oil over medium-high heat. Add as many pieces of beef as will fit in a single layer. Cook on all sides until the beef is well browned, about 8 minutes. Remove to a plate. Repeat with remaining batches, adding more oil to the pan if it seems dry. Do not crowd the pot, or the beef will not brown properly.
Reduce the heat to medium. Add the onion, celery, and mushrooms. Saute until softened, about 8 to 10 minutes. Adjust the heat as necessary so that the vegetables do not burn.
Stir in the garlic, tomato paste, Worcestershire, smoked paprika, and thyme. Cook, stirring constantly, for 1 minute.
Stir in the carrots and parsnips, coating them with the cooking oils and spices.
Return the beef to the pot. Pour in the stock. Add the bay leaf.
Increase the heat to medium high and bring the liquid to a simmer. Reduce the heat to low and partially cover the pot. Let simmer for 45 minutes, until the beef is fairly tender.
Stir in the barley and remaining ½ teaspoon black pepper. Continue to simmer over low, partially covered, until the barley is tender, about 45 minutes more.
Remove the bay leaves. Stir in the parsley and vinegar. If the soup is thicker than you would like, thin it with additional water as needed. Taste and adjust seasoning as desired.
TO STORE: Refrigerate soup in an airtight storage container for up to 3 days.
TO REHEAT: Rewarm leftovers in a Dutch oven on the stovetop over medium-low heat or in the microwave.
TO FREEZE: Freeze soup in an airtight freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.