Why my father always thinks that my mum have a boyfriend when she going out somewhere?

My dad.
He is anti social like our dad and acting like my mom is a slave that can't go out and enjoy her life. If never had been ONCE going out with friends, family, party drinking beer because my dad will mistaken that she have another boyfriend and making fool story like she went to see her boyfriend in the salon What the hell?
by the way my mum say she's going to break up with my dad but my dad says "Give me one more chance". He says it so often. He is having a mental problem? I don't know because he always thinks HE IS RIGHT. HE ALWAYS STAND FOR HIMSELF. But it's not true, He lies and his evidence is foolish. But he still say it's true. Hate him . He say he love my mom. But really?Also when are moving out of the house to another he say I apologize for everything I done then he keep repeating this Sh** again ,What shall I do?
Sometimes he is good sometimes he is bad.
I don't know whats going on with him?
He also call my mom a "B****"
I can't it stand anymore

Your dad sounds like he's very insecure, lacking in self confidence and unhappy with his life. If alcohol is involved, that will amplify the way he feels.

What you can do depends on your age. If you are still in school, it's all up to your mom. Consider whether she has enough income to live on her own. Talk to her when the emotional level is low and he's not around and tell her how you feel.

If you are old enough and capable of supporting yourself, look for someone else to stay. Maybe an aunt or grandparent could take you in in the meantime.

If there is no Trust in a Relationship, then there is nothing. If your Dad does'nt Trust your Mum then why are they together? You NEED Trust to make a Relationship work.