Preheat oven to 350ºF. Line 12 standard muffin cups with paper liners. In a large bowl, whisk together flour, cardamom, baking powder, and salt. In another bowl, whisk together brown sugar, eggs, oil, vanilla and the grated parsnip. Stir in flour mixture until combined.
Divide mixture evenly among lined cups. Bake until a toothpick inserted in the center of a cupcake comes out with only a few moist crumbs attached (*I always press gently on the center of the top - it should spring back), 18-20 minutes. Let cool completely in pan on a wire rack.
Make frosting: In a large bowl, beat cream chesse, butter, confectioner's sugar and vanilla until combined. Spread frosting on cooled cupcakes. (Frosted cupcakes can be refrigerated in an airtight container up to 2 days.)