Preheat oven to 180 degrees. Whisk together flour, cardamom, baking powder, and salt.
In a large bowl, whisk together brown sugar, eggs, oil, 2 teaspoons vanilla, and parsnip.
Stir in flour mixture.
Line 12 standard muffin cups with paper liners.
Divide batter among cups. Bake until a toothpick inserted in
center of a cake comes out with a few moist crumbs attached, 18 to 20 minutes.
Let cool completely in pan on a wire rack.
In a large bowl, with a mixer, beat cream cheese, butter, confectioners' sugar, and remaining teaspoon vanilla until combined.
Spread frosting onto cooled cupcakes