For the cake: Preheat the oven to 350 degrees F. Grease (with some vegetable oil) and flour a 9-by-13-inch cake pan.
In a medium saucepan, bring the parsnips and milk to a simmer over medium heat. Cook until tender, about 5 minutes. Remove from the heat, cover the pan and let stand until cool.
Drain the cooled parsnips, discard the milk and transfer them to a blender. Add 2 of the eggs and blend until very smooth. Transfer to a large bowl. Add the carrots, granulated sugar, salt, star anise, ginger, almond extract, white pepper, 3/4 cup vegetable oil and the remaining 2 eggs to the blender and blend on high speed until very smooth. Pour the carrot mixture into the pureed parsnips and stir well until combined.
Sift together the 2 cups flour and baking powder into a bowl. Add the dry ingredients to the wet and, using a large rubber spatula, fold together until well incorporated. Pour the batter into the prepared pan and bake, rotating the pan halfway through, until golden brown and a tester inserted into the center comes out clean, 45 to 50 minutes. Let cool completely on a cooling rack before removing the cake from the pan.
For the icing: Cream the cream cheese and butter together well. Add the confectioners' sugar and salt and mix well, making sure there are very few lumps. Combine the carrot juice with the vanilla, bitter almond and anise extracts and mix in a little at a time. Stir in the grated carrot last. Put back in the fridge to firm up. Frost the cake with the icing.