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We use "はは", "母", "ちち", "父" when you talk about own parents to someone in polite way.
ex) My father is fine. 私の父は、元気です。
We use "お母さん", "お父さん", "お袋", "おやじ" when we talk about them or call them.
Each one has different nuance, I think. I feel warm feeling from "お袋", "おやじ"。
However I never call like that for my parents.