Pour milk into a large pot over medium-low heat; bring to a simmer. Add parsnips and salt. Cover and cook until parsnips are tender, 25 to 30 minutes.
Drain parsnips, reserving warm milk in a separate container.
Mash together parsnips, 1 cup reserved warm milk, butter, thyme, and pepper in the same pot using a hand mixer or blender. Add more warm milk until desired consistency is reached. Season with salt and pepper.
