Actually, it doesn't mean anything at all. The sentence with the possessive doesn't make sense in Greek without the article. ._.
I think the sentence should be translated as "the parents of her", in order to understand the importance of the τούς. This is, it's showing the parents are the direct object
I think that, since there is especially on the sentence, there is no need for more emphasis, mostly because there is no hint of it. Even if there was, it would be more likely to say "Ειδικά για τους δικούς της γονείς", and that sentence would translate to "Especially for her own parents". And that's not the case here, so this sentence is fine just as it is. ^.^
"specifically for her parents" was marked wrong
isn't that an acceptable translation?