There's a myriad of ways to do this, but usually through slight of hand, he slipped you the duplicate card which you thought was the ripped up version - when it reality, was a folded version.
In the parent's description, it doesn't sound like he ever gave the card back to the magician. He himself ripped the card, then immediately keeping the pieces tight in his fist.
Right. The thing that blew our minds (my friends and I) was that I clenched my fist around the pieces and watched my hand like a hawk. The only rational possibility I see for misdirection would have happened before I knew I was supposed to close my hand; but hard to say. Greatest trick I've ever witnessed!
An easy way to do this is to have the folded card in your right hand (palmed), you place the ripped pieces in the same hand, but hold them with your fingers. You look like your folding the pieces up, then make the switch as you place it in his hand, fold it up and tell him to hold it tight.
Of course this is done in one fluid motion, so he doesn't see the slight of hand switch - like the guy does at the very end of the video.
Here's a great video of how this is done: http://bit.ly/yWQgQL