Heat 1 Tbsp olive oil in a Dutch oven or large stockpot over medium-high heat until shimmering. Add half of the beef in a single layer and cook until browned but not cooked through, about 2 minutes per side. Transfer to a plate. Repeat with browning the remaining beef, then transfer to the plate.
Heat the remaining 1 Tbsp olive oil in the same pot over medium-high heat until shimmering. Add the onions, carrots, celery, parsnip and turnips, and cook, stirring occasionally, until fragrant and begin to soften, 3 to 5 minutes. Add the garlic, 1 tsp dried thyme, and remaining 1 tsp kosher salt and cook until fragrant, about 30 seconds more.
Add 1 tbsp tomato paste and stir to coat the vegetables. Cook until darkened in color, about 1 minute. Return the beef and any accumulated juices to the pot and add beef stock broth, 3/4 cup pearled barley, 1 dried bay leaf. Stir to combine and bring to a boil over high heat. Reduce the heat to medium-low and simmer uncovered until the barley is tender, 40 to 50 minutes.
Remove and discard the bay leaf. Ladle the soup into bowls, garnish with the parsley, and serve.