I hesitate to pass judgment on anyone, especially a parent-child team. I know nothing about Everest except for what I've read, and I have no high altitude climbing experience.
That being said, I have two strong sentiments:
One: I feel that the vast majority of adults underestimate children. I suspect that most kids -- the ones that are growing up active and not sitting in front of the television set -- are physically capable of far more than their parents suspect.
Two: That being said, this is Everest, not Mt. Washington. I jokingly asked Hugh if he would allow Alex to do something like this while she's under the age of 18. His response was an immediate "hell no." I feel the same way. In my opinion, for my own family, the risk is far too great. I can see a Kilimanjaro attempt in our future, but that's the limit. She can do whatever she wants when she's older, but I don't want to take/allow unnecessary risks when it comes to my kid. (I am not comparing Alex's abilities to Romero's, just using her as a hypothetical to assess my own feelings on the matter).
However, that's just me, speaking for my own family. I can't pass judgment on somebody else's family. The kid certainly has enough experience and it's clear he's mature for his age and has a definite goal. I do hope the risk has been sufficiently explained to him, though. That's the only thing that wasn't clear in the article. However, he's 13 and not an idiot, so he has to know there are bodies all over that mountain...