Heat the oil in a large stockpot set over high heat until it shimmers. Add the tomato paste and stir constantly for 45 seconds. Add the stock and stir to combine. Add the barley, lentils, beets, carrots, and potatoes, stir gently, cover, and bring to a boil. Reduce the heat to a simmer and cook for 30 minutes, or until the barley and lentils are each partially cooked and slightly softened.
Add the salt, parsley, pickle juice, onion, celery, all but 2 teaspoons of the dill, the bay leaves, allspice, and black pepper, and stir. Cover and cook for 15 minutes, or until the carrots and celery are tender-crisp, the potatoes are soft when poked with a tip of a sharp knife, the barley is cooked through, and the lentils are beginning to break apart. Remove the bay leaves.
Ladle into bowls, garnish each bowl with 2 tablespoons diced pickle and reserved fresh dill, and serve immediately.