All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Anger is dangerous, and it is one of the most powerful weapons of the devil which he uses to realize his objectives in misguiding the sons of Aadam, and this is why the Prophet warned against it. It was narrated on the authority of Abu Hurayrah that a man said to the Prophet "'Give me advice.' He said: 'Do not get angry.' The man repeated the same request several times, and each time, he gave the same advice: 'Do not get angry.'" [Al-Bukhaari]
If your father gets angry often, then you should avoid whatever may provoke his anger as much as possible. He should also be advised and reminded of the danger of anger and its bad effects on a person’s religiosity and worldly life. It is better that the person offering the advice should be someone who is hoped to have an influence on him from his relatives or otherwise, and the advice should be given kindly and gently. Such kindness and gentleness are required for greater reason if the advice is given by any of his sons or daughters. Allah, The Exalted, Says (what means): {And lower to them the wing of humility out of mercy and say, “My Lord, have mercy upon them as they brought me up [when I was] small.”} [Quran 17:24]
Raising your voice with him or speaking to him in the manner described in the question is impermissible, and it is rather a kind of undutifulness towards him. It is incumbent on you to repent of it and seek his forgiveness. If you remark on his wrong actions in a polite manner within the limits of the Islamic etiquette, then there is nothing wrong with that, and it is not considered undutifulness towards him.
As his children, it is obligatory on you to show dutifulness and kindness towards him. His offense against you, regardless of its extent, does not justify offending him, nor does it waive the obligation of showing kindness to him. Allah, The Exalted, Says (what means): {But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them but accompany them in [this] world with appropriate kindness…} [Quran 31:15]
Your mother is not allowed to speak harshly and angrily to your father but should rather advise him kindly and gently. In general, advice should be given kindly and gently by any person, and this requirement is especially emphasized with regard to a wife towards her husband given the great rights he has over her. The Prophet said: "Were I to command anyone to prostrate before anyone besides Allah, I would have commanded the wife to prostrate before her husband given the great rights that Allah has given him over her." [Abu Daawood]
On the other hand, your father has no right to insult your mother, for he is required to live with her in kindness. Allah, The Exalted, Says (what means): {And due to the wives is similar to what is expected of them, according to what is reasonable. But the men have a degree over them [in responsibility and authority]. And Allah is Exalted in Might and Wise.} [Quran 2:228]
It is noteworthy that disciplining one’s children must not be done out of revenge or aggression. When needed, parents should discipline their children within the limits of the Sharee‘ah guidelines.
Allah Knows best.