Make two 8-inch-round chocolate cakes as the cake mix label directs; cool completely. Crumble 1 cake into a bowl. Squeeze a handful of crumbs into a 2-inch ball; roll between your hands until smooth. Repeat to make 2 more cake balls.
Sprinkle some of the remaining cake crumbs on a clean surface. One at a time, roll the cake balls in the crumbs, pressing lightly to coat. If necessary, moisten the cake balls with water to help the crumbs stick.
Assemble the cake: Put the remaining whole cake on a serving plate or cake board. Spread about 1 1/2 cups vanilla frosting on the cake with an offset spatula; run the spatula around the sides to make it smooth.
Arrange the cake balls on the cake. Mix the remaining frosting, the yellow food coloring and cocoa in a bowl; transfer to a pastry bag with a small round tip. Pipe the yellow frosting around the cake balls in a figure-eight motion to look like spaghetti.
Grate white chocolate on top of the cake to look like parmesan cheese. Serve the cake with the remaining strawberry puree on the side.
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