You punish kids because their parents are cretins? If/when the parent asked me about bringing an extra child, I would explain that there are a set number of places in games, gift bags, etc., and warn the parent that the uninvited child will have to go without. This might move them to take the child to the movies or something with a special treat afterwards.

As an aside, I've had the cake pictured in ithe photo. It is TREMENDOUS.