I read the definition of hypochondria, but it does not say: is it correct to use this word, for instance, about parents who worry too much about their kid(s) being sick?
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Hypochondria does indeed refer to the sufferer himself.
Worrying about other people's health should be called hypochondria by proxy (by extension from Munchausen by proxy) but I can't find a reliable reference for that. The best one is a blog which implies that a vet has diagnosed it in a pet's owner.
There does appear to be a defined Hypochondria by proxy. This medical journal article even discusses two cases in 1999: