grammaticality - "both of" + possessive, which noun does "both of" refer to?

Both of the boy's parents were happy with the new school.

Is it proper English to say "both of the boy's parents", as in the above sentence, to mean "both parents of the boy"? Or do we have to use the latter?

Background

A similar question appeared on a private advanced-level English test. The task was to spot a mistake in a sentence similar to the above. The correct answer was apparently the place of the apostrophe: in terms of the above example, it should read "both of the boys' parents" (meaning "parents of both boys"), with the argument that, in proper English, "both" can only refer to "boys" in such circumstances. I found this argument a bit shaky though. I am not a native English speaker, but I have studied and spoken it for many years, and the above sentence looks perfectly correct to me. I also could not find any helpful references on the internet that address this scenario.

Answer

I would say the sentence is perfectly fine with boy's, and the test is mistaken. There is absolutely nothing wrong with both of the boy's parents. The only reason I can think of why it would be wrong is if context made it clear that this had to be about two boys. That is possible, if this sentence is part of a story; was it?