Polite ways to refer to somone's parents in Arabic
The word أٌمُ means mother in Arabic, and the word أّبُ means father. And we know that the attached pronoun ك is used to talk about someone else's posession. Like أٌمّك (your mother), and أُبُوك (your father).
While it's not wrong to use these words, there are certain cultural sensitivities to keep in mind when using these two words.
Sadly, for certain people it's associated with some insults that some people who say (like cursing someone's dad or mom), sometimes even the word أمك or أبوك when said angrily on its own can imply an insult.
In many parts and regions, many Arabic speakers would even avoid or refuse saying their mom or sister's names entirely because they don't want others to use their names and insult them. Some people overdo it and it can be a bit over the top with this overprotective attitude, but people are now becoming more aware that they're overdoing it.
This reminds me of a video where a guy was talking to his Arabic speaking friend, trying to say something sweet like "say hi to your mother" or "your mother is beautiful" or "your mother is awesome" but accidentally offending his friend and earning wit wrath instead of his gratitude.
For some, saying أمك or أبوك can be a bit direct, not necessarily rude, but some people won't really like it especially if the speaker is emotioanl or angry. In this post, I'll teach you here some alternatives to say in order to avoid sounding too direct :
الوَالِد (for the father) and الوَالِدة (for the mother)
It literally means "the male parent" and "the female parent". It's really polite, you can say it so someone you're not close with or to someone you're trying to be formal with. Some guys however still say الوالد and الوالدة out of politeness. The guy in the video could have avoided an anger fit if he has said سَلِّملِي على الوالدة (sallimlī ʿal wālda; say hello to your mother) instead of saying (sallimlī ʿalā ommak).
Or if you want to talk to your colleague or boss about his mother who is very sick or something
أَلْف سَلَامَة للوَالِدة (alf salāmeh lal wālda) which means something along the line for "I hope your mother gets well soon"
خَالتُو (auntie) or عَمّو (uncle)
The word خَالة means the maternal aunt (your mom's sister) and the word عم means the paternal uncle (your dad's brother), but people use it to say "uncle" or "auntie". You can say it to your friends in reference to their parents in situations where the word الوالد and الوالدة sound too formal
كِيف خالتو؟ طمني عليها/ "kīf ẖalto? tamminnī ʿaleiha"/ How's auntie? Reassure me (that she's ok)
الحاجّ (masc) or الحاجة (fem)
The word حَاج means someone who performed pilgrimage, and the word حاجة is the feminin version, although they are usually pronounced as حج and حجة (replacing the long vowel ا with its short equivalent, fatha) in dialects .
It's polite and respectful to an old person to use these words, kinda like using "granny" in an endearing way to your friend's grandma or to your old neighbour (either when speaking to them or referring to them).
كِيف الحجة؟ طمني عليها/ "kīf elḥajjeh? tamminnī ʿaleiha"/ How's grannie? Reassure me (that she's ok)
These words are used for elderly people, middle aged people might be sad you're referring to them as elderly (it's not rude, but they might feel that you see them as elderly, as in too old). Perferably use it to refer to people's old grandparents. For parents you can use the options above.
Other ways to say the name
In Arabic, there is a certain concept that is called كُنِيَة (kunya in MSA), or (kinyeh in dialects)
It's a commonly used nickname for the parents, they usually add أم (for the mother) and أب (for the father) followed by the name of the eldest son.
For example if a man has 5 children, the eldest son is called Ali (even if he wasn't the oldest child, as in he has older sisters), the man's كنية would be "Abu Ali أبو علي " and his wife's كنية would be Um Ali أم علي .
In more formal setting, Arabic speakers would usually avoid using the first name of the parents and prefer to use the كنية, some people would find it too formal and would rather use خالتو or عمو (uncle / auntie), it depends on the person and what kind of relationship you have with the person.
Other options
If you're talking to a child for example, like a yonung relative or even a lost kid at the mall, you can say مَامَا (mama) or some would even say مَامتك (your mommy).
For example if you're asking the son of your friend about your friend you could say
كِيف الماما؟ سَلِّملِي عَليها / keif el māmā? sallemli ʿaleiha / how's your mama? say hi to her.
It's ok to use to a child, but if you say it to an adult for example it would kinda sound like you're teasing them or telling them that they're childish in a way.
Which word to choose?
These are observations. Of course, it doesn't mean that every single person finds the words أمك or أبوك to be too direct but still, it's good to keep this idea at the back of your mind.
My suggestion is to first of all look at your relationship to that person, and to see how that person refers to their own mother and to other people's mothers.
Some people would refer to their own moms as الوالدة, this could be your queue to refer to their mom like that as well and so on.
The new younger generation are not as sensitive to these things, but some people would still find it too direct.
The point of this post is to make you aware of these differences, so that you won't be in an embarassing situation.