To prepare cake, preheat oven to 350°.
Place the parsnip in a saucepan; cover with water. Bring to a boil, reduce heat, and simmer over medium heat 20 minutes or until tender. Drain.
Place the parsnip, 1 cup brown sugar, granulated sugar, 5 tablespoons butter, and 1 teaspoon vanilla in a food processor, and process until smooth, scraping sides of bowl occasionally. With the processor on, slowly add apple juice through the food chute, processing until well combined. Place the parsnip mixture in a large bowl.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking soda, cinnamon, nutmeg, and 1/2 teaspoon salt, stirring well with a whisk. Add flour mixture and 3/4 cup milk alternately to parsnip mixture, beginning and ending with flour mixture. Pour batter into 2 (9-inch) round cake pans lightly coated with cooking spray. Bake at 350° for 30 minutes or until a wooden pick inserted in center comes out clean. Cool in pans 10 minutes on a wire rack; remove from pans. Cool completely on wire rack.
To prepare icing, combine 1 cup brown sugar, 1/2 cup milk, 2 tablespoons butter, and 1/4 teaspoon salt in a medium saucepan. Bring to a boil over medium-high heat, stirring constantly. Reduce heat, and simmer until slightly thick (about 5 minutes), stirring occasionally. Remove from heat. Add powdered sugar and 1/2 teaspoon vanilla; beat with a mixer at medium speed until icing is smooth and only slightly warm. (Icing will continue to thicken as it cools.)
Place 1 cake layer on a plate. Working quickly, spread the layer with half of the icing, and top with the remaining layer. Spread the remaining icing over the top layer, and sprinkle with toasted pecans. Cool to room temperature. Store cake loosely covered in the refrigerator.