More than an afterthought on a crudité platter, celery is the star in this simple, silky soup. Slideshow: More Soup Recipes
In a large saucepan, melt the butter over moderate heat. Add the onions, garlic and a generous pinch each of salt and pepper; cover and cook, stirring occasionally, until the onions are softened, 8 to 10 minutes. Add the celery root and potato and season generously with salt and pepper; cover and cook, stirring occasionally, until tender, about 12 minutes. Add the sliced celery, cover and cook, stirring occasionally, until just tender, about 7 minutes.
Add the chicken broth to the saucepan, cover and bring to a boil over moderate heat. Reduce the heat to moderately low and simmer, partially covered, until slightly reduced, about 15 minutes.
Thinly slice the darker green celery leaves, reserving the smaller inner leaves for garnish. Add the sliced celery leaves and the parsley to the soup and simmer until wilted, about 2 minutes.
Working in batches, puree the soup in a blender until smooth. Transfer the soup to a clean saucepan, rewarm over low heat and season with salt and pepper. Ladle the soup into bowls, garnish with the reserved celery leaves and a drizzle of olive oil and serve.
The celery soup can be refrigerated for 4 days.