Heat the chicken stock in a pot to a gentle simmer.
Heat the vegetable oil in a large pot over medium heat and saute the onion until translucent. Add the pearl barley to the pot and stir for one minute. Stir in the hot chicken stock, turmeric, lime juice, tomato paste, salt, and pepper. Bring the mixture to a boil, reduce heat to low, and simmer for 1 hour.
Mix in the carrots and continue simmering 30 minutes or until the soup has thickened and the carrots and barley are tender. If the soup is too thick, add hot water, one tablespoon at a time.
Place the sour cream in a small bowl. Slowly pour 1/2 cup of hot soup mixture into sour cream, whisking constantly. Gradually add the sour cream mixture into the soup pot, whisking constantly. Stir in the fresh parsley.