Put the celery root and parsnips into a medium pot and cover them with the milk. Bring to a simmer and let cook until the parsnips and celery root are tender and cooked through, about 30 minutes. Strain the vegetables from the pot, making sure to reserve 1 cup of the milk before discarding the remainder. Return the parsnips and celery root to the pot, add the butter and mash the ingredients with a potato masher. If you want a smoother texture, then blend in a food processor.
Add in some of the reserved milk to lighten the consistency. Adjust the seasonings with salt and pepper and stir until well incorporated. Transfer to a serving bowl and serve.