Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 10-cup fluted tube pan.
Combine cake mix, pudding mix, eggs, sour cream, butter, water, and sugar in a mixing bowl. Beat on low speed until ingredients are moistened, then beat on medium-high speed for 1 minute. Batter will be thick. Divide batter evenly into 6 bowls. Tint each batch a different color, using enough food coloring paste to reach desired tone; colors will deepen and brighten when baked.
Spoon red batter into the bottom of the prepared pan and smooth using an offset spatula to evenly distribute. Carefully spoon purple batter on top and spread with spatula, pressing batter thinner near the inner edge and thicker at the outer edge of pan to create an arc effect. Repeat with blue, green, yellow, and orange batters. Gently tap pan on the counter 3 times to allow batter to settle.
Bake on the middle rack of the preheated oven until a toothpick inserted in the center comes out clean and top is just starting to brown, 35 to 45 minutes. Be careful not to overbake.
Cool cake in the pan for 10 minutes. Turn cake out onto a plate and allow to cool completely before frosting, about 1 hour.
Drizzle or spread frosting over cake and decorate with rainbow sprinkles.
