In a medium saucepan, bring the stock, tomatoes, basil, lovage, and parsley to a boil. Lower the heat and simmer until the tomatoes are very soft, about 15 minutes.
In a food processor, purée the soup, then pass through a fine-mesh sieve; or pass through the fine disc of a food mill. The soup can be made ahead up to this point and refrigerated for 2 days.
Scrape the soup back in