Put parsnips in a pot, season with salt and cover with milk and cream. Add garlic, bay and thyme and bring to a simmer over medium heat. Cook until tender, the tip of a paring knife should easily go through without resistance, about 12 to 15 minutes. Strain and reserve liquid.
Place parsnips in a food processor with butter and pour in enough of the reserved milk mixture to puree the parsnips and achieve the texture of whipped cream. Season with salt and pepper then finish with a sprinkle of parsley.