Is the wife obligated to re-pay the zakat on gold that she gave to her parents, given that she did not know it was impermissible to do so?

If the wife is unable to support her parents, or if she paid zakat for a debt they owed, then the zakat is valid. However, if supporting her parents was obligatory upon her, and she paid zakat to them, being unaware of the ruling, then there are two opinions regarding the validity of the zakat. The more cautious approach is to pay the zakat again. She has the right to reclaim from her parents what she paid if they knew it was from zakat.