Does he have to pay zakaah on money which his father is saving for him?
Praise be to Allaah.
The obligation of zakaah is subject to several conditions:
Freedom: zakaah is not obligatory upon a slave, because he has no property or wealth. Whatever he owns is the property of his master, so the zakaah is obligatory upon his master.
Islam: because the Prophet (peace and blessings of Allaah be upon him) said, according to the hadeeth of Mu’aadh (may Allaah be pleased with him): “Call them to testify that there is no god except Allaah and that Muhammad is the Messenger of Allaah,” then he mentioned prayer, then he said: “and if they obey you in that, then teach them that Allaah has enjoined upon them charity (zakaah) to be taken from their rich and given to their poor.” (Narrated by al-Bukhaari, al-Zakaah, 1365; Muslim, al-Eemaan, 28).
Owning the nisaab (minimum amount or threshold): zakaah is not obligatory upon wealth or property that does not meet the minimum amount, which is a known amount of wealth.
Full control: such that no one else has any rights to the wealth; there is no zakaah on wealth that one does not own with full control.
The passing of one complete (hijri) year: because of the hadeeth of ‘Aa’ishah (may Allaah be pleased with her), who said: “I heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say: ‘There is no zakaah on wealth until one full year has passed.’” (Narrated by Ibn Maajah, al-Zakaah, 1782; classed as saheeh by al-Albaani in Saheeh Sunan Ibn Maajah, no. 1449).
These are the conditions for zakaah being obligatory; if these conditions are met, then zakaah is obligatory.
Based on the above, it may be said that you do not possess the money, let alone saying that you have full control over it. If you possess the money and the other conditions are also met, then you should pay zakaah on it. And Allaah knows best.
For more information on the conditions of zakaah, see al-Mulakhkhas al-Fiqhi by al-Fawzaan, 1/221; al-Sharh al-Mumti’ by Ibn ‘Uthaymeen, 6/22