Assalamu alaikom wa rahmatu'llahi wa barakatuhu,
I agree for mother priority over the father. And also, the use of violence against her, you father was totally wrong. There is not one story or hadith about Muhammad Sallah Allahu Alaihi Wa Sallam hitting one of his wives nor anyone in his entire life except in Jihad.
And for that, you was wrong in even pushing your father ! I think if you pushed you father, you could normally stop him with pushing by just stand in middle (and maybe get hit) I know Islam calls for defending ourselves but we are talking about the father here. I really think you could find a workaround pushing him as this could deeply affect you relation, be patient with your father and pray Allah Subhanahu Wa Taala so he leads your father.
You must apologize to your father for pushing him, and tell him that was not intentional even if he was wrong.
As mentioned in the Coran, the husband should take a series of actions when his wife deviate from Allah Subhanahu Wa Taala's path, when she does something wrong (something very important which affect her religion or disobey her husband...).
The first thing to do when a situation occurs is to talk to the wife, the husband can do it until she understands and how long it will take until that, depend on the husband patience.
Second solution, as mentioned in Coran, is to seperate from the wife in the bed (I hope you understand this because it is very important and a lot of muslims get it wrong) : no one in the family should notice the the husband is not touching his wife, the husband must not quit the sleeping nor even the bed. He should only avoid his wife. Why is that ? Because in avoiding her (only in the bed), he breaks her pride. She will certainly ask herself why her husband is not touching her when Allah Subhanahu Wa Taala gave him the right to ? My brother and Sisters, Allah Subhanahu Wa Taala created us and told us how to live and how to act in every situation, if HE told us this second solution, that means it has its effect on the wife.
Third solution, comes the "hitting"; The verb to hit in arabic is يضرب / yadhreb can mean a lot of things, and one of the usage of it can be translated (word by word in arabic) like this : He hits an exemple to explain the students ! (Sorry for my poor english !). And there is a very good interpretation of the Aya in the Coran about the hitting, to hit can mean " to make an impact " and this impact can be moral or physical so before the physical impact, one should try the moral one ! (not insults, just serious words to explain the problem...) As for the physical impact, Rasulu'l Allah Salla Allahu Alaihi Wa Sallam ordered muslims, when they hit someone, to not break bones nor make someone bleeds nor make the skin change its color.
So your father shouldn't beat your mother, he was wrong.
and you defended your mother which is the right thing to do.
but you shouldn't push you father, you was wrong. When you will have kids Inshaa Allah and raise them, and hold them in your arms, get sleepless nights for watching over them until they grow up, and if one day you are wrong about something to the point your own kid hit you, wont you think and remember those old days when your kid were not even able to eat by himself.
My Allah Subhanahu Wa Taalah leads your family to real happiness and to the right path, indeed HE is able to do it, He is the Most Powerful. Ammen.
Assalamu alaikom wa rahmatu'llahi wa barakatuhu.
