Is a son's supplication against his father—if the father is a liar or mentally ill—and God's answering of that supplication by making the father bedridden, considered a sin upon the son?

It is impermissible to supplicate against one's father, even if he is unjust, as this constitutes undutifulness (Uquq), which is one of the gravest major sins. Sheikh Ibn Baz was asked about the ruling on supplicating against an unjust father, and he replied that it is impermissible. Instead, one should say: "O Allah, guide him. O Allah, protect us from his evil. Allah is sufficient for us, and He is the best Disposer of affairs." The fundamental principle is to treat parents kindly, even if they are disbelievers who invite to polytheism. One must supplicate for his guidance, the rectification of his intentions and actions, and that Allah protect from his evil. What you did by supplicating against your father is a sin from which you must repent. What is obligatory is to treat him kindly, advise him, and supplicate for his guidance and uprightness.