Heat oil in a large saucepan over medium heat until shimmering. Add onion and cook, stirring occasionally, until soft and starting to brown, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds. Add parsnip, celery root, potato and 6 cups liquid and bring to a boil. Reduce heat to a simmer and cook uncovered until vegetables are soft, about 15 minutes. Remove from heat and allow to cool for a few minutes.
Puree soup either by transferring to a blender (do not fill blender more than 1/2 full and process in batches) or puree in the saucepan using an immersion blender. Adjust the thickness by adding an additional 1-2 cups of liquid and mix to combine. Add the cream, sugar, vinegar, salt and sugar and adjust seasoning as needed.
To serve, portion soup into individual bowls and drizzle with truffle oil, toasted bread crumbs and a few leaves of parsley. Serve.