Place the parsnips and potatoes into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 20 minutes. Drain and allow to steam dry for a minute or two.
Place the vegetables into a food processor, and process until smooth. Add the milk and horseradish; season to taste with salt and pepper. Process until mixed.