The amount of vicious comments in this thread is fairly disheartening.
I'm inclined to believe the kid's story. It was mentioned in the article that earlier payments were agreed upon between the mother and the child, making it more likely that the kid has a relatively good understanding of the consequences of making in-app purchases.
Does that make it necessarily wise to give the kid full access to the card? Arguably not, but even so the child's version of events seems very plausible to me.
Also, this is easily a mistake an adult can make. I'm willing to bet many, many people here have had experiences with adult computer users who, for example, click "Print" or their Internet Explorer icon multiple times when it doesn't initially print or open up, just to have 40 copies print or 40 instances of IE all open at once at them.
But by all means judge her credibility as a parent based on this one snapshot into her life and laugh at her potentially devastating financial loss (I know I'd be pretty fucked if I lost 3k out of my bank account in short order).