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 and especially have quite different meanings to my ears. "Specifically", in this case would highlight that "it" is definitely only for the parents, while "Especially" could mean that "it" is not just for the parents, but they are entitled to more of "it" than others.