why not make a letter with your full name, address and contact number giving permission for your father [ full name, address etc] to take your daughter to Aruba, he can keep the letter just in case the authorities ask him does he have your permission, if he doesnt need show it all well & good, but if asked he has all the imformation and your signature and contact details to prove it is all legal & above board.
I took my 6yr old grandughter to Orlando. Florida [ ages okay] with my then new husband, I had not changed my passport so we were travelling in three different names, the US immigration questioned her in a roundabout about way as they searched our bags on arrival, my grandaughter piped up " excuse me but you are being rather nosey" and the guy laughed and said go, but at that time we did not think to bring a permission letter, we were grandparents taking our grandaughter to Disney.Times have changed now.