To make candied parsnips, combine water and 1/4 cup sugar in small saucepan. Bring to a boil, stirring occasionally until sugar dissolves. Reduce heat to low and add parsnips. Cook for 5-6 minutes or until parsnips are tender; stirring occasionally. Remove from heat and strain.
In small bowl, toss remaining sugar and cooked parsnips together. Spread in an even layer over parchment paper to cool and dry.
For cake, combine flour, granulated sugar, brown sugar, baking powder, salt, nutmeg and ginger in small bowl.
In large bowl, whisk eggs until foamy, about 1 minute; add oil in a thin stream and whisk until well combined. Stir in flour mixture until just combined. Stir in parsnip and nuts. Divide evenly into prepared pan.
Bake 22-24 minutes or until toothpick inserted in center comes out clean. Cool 5 minutes in pan. Remove from pan to cooling grid; cool completely.
For glaze, whisk together confectioners? sugar, browned butter and salt in small bowl. Add milk 1 tablespoon at a time until thick but spreadable. Ice tops of cakes; immediately sprinkle with candied parsnips.