I would say horrified but that's just because the word horrified is more common in my opinion. Aghast is less common and quite strong.

In this case they're synonyms. Both are used and both would be correct. They achieve the same outcome; he is not happy about his son doing it. I would say that perhaps aghast implies that he's taken back, or shocked at the idea. Whereas horrified suggests that he is scared of what others might think. But they are slightly subtle interpretations at a level that no one would really consider when reading/hearing it. Really, there's not a lot of difference in this case.

Horrified is more commonly used in literature and conversation.