Heat chicken stock in a large saucepan to a gentle simmer.

Heat vegetable oil in a large pot over medium heat. Add onion; sauté until translucent. Add barley; stir 1 minute. Stir in hot stock, tomato paste, lime juice, turmeric, salt, and pepper; bring to a boil, then reduce heat to low, and simmer 1 hour.

Stir in carrots; continue simmering until soup thickened and carrots and barley tender, about 30 minutes. If soup is too thick, add hot water, 1 tablespoon at a time.

Slowly pour 1/2 cup hot soup into sour cream in a bowl, whisking constantly; gradually add to soup, whisking constantly. Stir in parsley. Serve with fresh lime wedges.