A Jim Steinmeyer puzzle Christmas card

This is an example of a so-called ‘vanish’ puzzle. A simple version would be to have 8 poles printed on three pieces of card. When you swap the position of the two smaller pieces, and count again, there are only 7 poles. The puzzle is: where has the eighth pole gone? Jim Steinmeyer created this more complicated version for a Christmas card which he and Frankie sent in 2002. I include it here with his kind permission. Print it out, cut it into three pieces and try and figure out how one of the gifts becomes magically wrapped.