Father, dear Father, that’s a capital idea.
Father is a noun like any other, except when it’s also a name. The same goes for mother, aunt and uncle. There are times when these take a capital and times when they don’t.
When you use Father as a name – For goodness’ sake, Father, must you smoke that pipe in here? – it takes a capital. You could replace it with George in that instance (unless his name isn’t George).
My father likes to smoke his pipe in the kitchen. In this case, I’m saying that the man who stands in relation to me as a father likes to smoke his pipe there.
Mother objects when Father smokes his pipe in the kitchen. Dora objects when George smokes his pipe in the kitchen.
My mother objects when my father smokes his pipe in the kitchen. You can’t replace the words mother and father with Dora and George in this sentence – my Dora and my George; therefore they are not standing in for names and take no capital. Try replacing them: My parents argue when Father smokes his pipe. You wouldn’t write my Parents.
Does your mother object when your father smokes his pipe in the kitchen? You could replace mother and father here, but it depends on context. If I know your parents as George and Dora, I could ask, Does Dora object… but as written, I’m referring to them indirectly through their relationship to you, as before.
You can of course apply the same rules to Mum and Dad.
Part of the perceived problem arises because these two people are your parents. If you changed the words, you’d see the difference.
My brother likes to smoke his pipe in the kitchen. My sister likes to smoke her pipe in the kitchen. (We’re an equal opportunities blog here.) You would probably not in modern parlance refer to your siblings as Brother or Sister: For goodness’ sake, Sister, must you smoke your pipe in here? That sounds rather Victorian. You would be more likely to refer to her by name instead. For goodness’ sake, Penelope…
You would capitalise Sister, however, if the lady you refer to is a nun or a senior nurse in charge of a ward. In those cases Sister is a title. Brother also can be a title if referring to a member of an order of monks; similarly, there is Father (a priest), Reverend Mother, and Mother Superior. There is also Father of the House (parliamentary), which again is a title.
My aunt objects when my uncle smokes his pipe anywhere at all. Aunt objects when Uncle smokes his pipe. Aunt May objects when Uncle Simon smokes his pipe. Uncle Simon carries on regardless.
To sum up: if you could replace the word with a name, use a capital. If it’s a title, use a capital. If neither of these applies, don’t.