Melt butter in a Dutch oven or medium heavy-bottomed pot over medium heat. Add parsley root and leeks; cook, stirring occasionally, until leeks are just tender, about 7 minutes. Add potatoes, stock, and 1 cup water; season with salt. Raise heat to medium-high; cook until parsley root is tender, about 15 minutes.
Let cool slightly, about 10 minutes. Working in batches (so the blender is never more than halfway full), puree soup until completely smooth.
Return soup to pot; add milk. Heat over medium heat, stirring occasionally, until hot (do not let simmer), about 7 minutes. If desired, add more milk for a thinner consistency.
Serve soup with Crabmeat Toasts.