Hm, can't say I like the parents much from their comments, BUT I think that part of the problem is caused by the Tate Modern itself (and other museums/galleries).
If you position yourself as a family-friendly gallery, you offer art Saturdays, kids' activities, a restaurant advertised as family-friendly and a soft play area, you have to expect children to visit. The problem with having a soft play area in the middle of a gallery, and interactive exhibits, is that whilst it's obvious to most adults as to which bits of brightly coloured plastic are meant to be climbed on and which bits are meant to be admired respectfully from a distance, it's not that obvious to a child - even a nine-year-old.
I go to the Tate Modern a fair bit, both in a work capacity and with my DC for their entertainment (though they're only 4 and 2), and it really is hard to explain to them what makes one thing art and another cheap stuff to be played with. I do think a 9yo should know better, and yes, the parents/carers should supervise better, and it is bloody hard to find any sympathy whatsoever for the smug parents (why bother with the anti-establishment bullshit, why not just say the child was tired, bored and hungry and there was no obvious barrier to stop her...) but museums/galleries need to think about how to reconcile the need to attract younger audiences with the need to protect the works of art. Most family-friendly museums put rope barriers around anything remotely climbable.