Is it permissible to delay waking one's father for prayer—while attempting to awaken him through indirect means—to avoid his stubbornness and annoyance, or is it obligatory to wake him directly?

Waking a sleeper for prayer before its time is a matter of difference among scholars. Some say it is obligatory (Hanbalis), while others say it is recommended (the majority). You should inform your father that you will wake him for Fajr prayer, and try to wake him when the time for prayer begins. If he wakes up, then praise be to Allah. If he does not, then you have fulfilled your duty. If you do not attempt to wake him at all, there is no sin upon you according to the majority of scholars.