If the red is a solid shape beneath the white and you want it to appear that the white has cut the parts out of the red, then you could group the red and white shapes and go to Effect > Pathfinder > Subtract. As an effect, this method is non-destructive.
Another option would be to select all of the parts and use the Shape Builder Tool (Shift + M). Hold down the Alt key and drag over the white parts to permanently remove them from the shape.
Selecting the entire image then using Compound Path did the trick after all.
Are these all Illustrator shapes?
In that case try the pathfinder panel or the shapebuilder tool.