Bring a large pot of lightly salted water to a simmer, add celery root pieces, and simmer until soft, about 30 minutes. Drain.

Place celery root in a blender and process until smooth. Transfer puree to a pot, add vegetable broth, and bring to a boil over medium-high heat. Reduce heat and simmer for 5 minutes. Add cream and cook until hot, but don't boil. Season with salt, pepper, and nutmeg.

Meanwhile, melt butter in a skillet over medium heat and toast bread cubes until dry and lightly browned, 3 to 5 minutes.

Ladle soup into bowls and garnish with chopped parsley and croutons.