Preheat oven to 350º F. In a medium bowl, whisk together flour, nutmeg, cinnamon, cloves, baking powder, salt.
In another bowl, whisk together brown sugar, eggs, oil, vanilla and parsnip. Stir in flour mixture until well blended.
Line 10 standard muffin cups. Divide batter among cups. Bake 18 to 20 minutes. Let cool on a wire rack.
In a large bowl, make either frosting and spread onto cooked cupcakes.