All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is not obligatory on either of the parents to pay pocket money to the children, but the father must spend on his children who do not have money. It is the father who is obliged to spend on his children and not the mother. The spending is measured based on what is sufficient for food and drink.
Ibn Qudaamah said:
"Ibn Al-Munthir said: 'The scholars unanimously agreed that one must spend on his children who have no money [..]. The obligation with regard to spending on a relative is to spend what suffices them because it became an obligation because of their need. So, it is measured based on the amount by which the need is fulfilled.'" [End of quote]
If the father did so, then he did what he is obliged to do.
If he wishes, he may give his child some money as a gift in addition to what he spends on him, but he is not obliged to do so as we have already mentioned.
However, if he gifts one of his children, then he has to be just by gifting the same to the other children, and he should not favor any of them with a gift except for a sound Islamic justification.
Allah knows best.