What should I do if my brother asked me not to tell our parents about a doctor's appointment that our father knows about, but is unaware of its time, and then my father asked me about the time of the appointment? Is this considered a secret that I am not allowed to reveal, and must I use a veiled statement or only tell the truth?

You must guard others' secrets and not disclose them, for Allah Almighty says: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]," and the Messenger of Allah (peace be upon him) said: "When a man relates a narration and then looks around, it is a trust."

Therefore, if your brother asks you to keep something secret, it is not permissible for you to tell your father about it, unless there is a legitimate interest that necessitates it, such as fearing that keeping it secret would harm your brother.

You can use 'tawriyah' (ambiguity/double meaning) when necessary. For example, you could say: "Your brother did not tell me," meaning at this particular moment, and do not tell him that he forbade you, as it might sadden him or lead to adverse consequences.