Melt butter in a large pot over medium-low heat. Add onion and salt. Cook, stirring occasionally, until onion has softened and turned translucent, about 10 minutes.
Stir garlic into the onion mixture; cook until fragrant, about 1 minute. Add chicken broth and parsnips; bring to a boil. Reduce heat to a steady simmer; cook until parsnips are very tender, about 20 minutes. Puree soup with a hand-held blender until smooth.
Pour milk into the soup and heat gently, but do not boil, about 4 minutes. Season with salt and pepper.
Vegetable broth can be substituted for the chicken broth. Use whole milk instead of low-fat if preferred.
Leek can be used in place of onion if desired.
You can substitute white pepper for black pepper if desired.
Instead of a hand blender, you can use a blender or food processor to whirl the soup in batches. Cover blender with a kitchen towel to avoid potential burns. Strain soup through a fine-mesh sieve for an ultra-smooth soup.