Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-inch pan.

Mix together cake mix, 1 box pudding mix, eggs, oil, food coloring, water, vinegar, and cocoa. Pour into prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 35 to 40 minutes. Let cool completely.

To make the frosting: Beat milk and cream cheese together until smooth. Beat in pudding mix until combined. Fold in whipped topping until no streaks remain. Spread frosting over the cooled cake. Refrigerate until ready to serve.