Heat olive oil in a large heavy stockpot over medium heat; cook and stir onion, celery, and garlic in hot oil until onion is slightly tender, about 8 minutes.
Stir chicken stock, chicken, carrots, parsnips, and barley into onion mixture; bring to a boil. Reduce heat and simmer until barley is slightly tender, about 30 minutes.
Stir sweet potatoes into soup and simmer until sweet potatoes and barley are tender, about 30 minutes; season with dill, salt, and pepper.