No, I think that mzazi specifically means 'parent'
And, correct me if I'm wrong, but I'm 86% sure that mzazi isn't commonly used to describe 'father' or 'mother'. But, since Swahili is an African language, nor European, Asian or American (which I specialize in), I'm not sure.
Hope this helps!
EDIT: ngwarai gave me a clarification, yes it does mean parent.
Source: Native Bantu speaker
That's fine. The question, though, is whether you would ever refer to just one parent. In English, I hear the plural "parents" all the time, but I never hear the singular, because if an English speaker is referring to just one, he nearly always specifies either "mother" or "father." My question is whether in Swahili you would commonly refer to one parent as "mzazi wangu," or whether you would specify "mama yangu" or "baba yangu," as we would in English.
Yes you can refer to just one parent as "mzazi", either yourself or another person saying or asking. You can also use "baba" or "mama" There are situations where either is more appropriate than the other.
I will give you one example; When you are misbehaving to your mother and she is angry with you, she will most certainly remind you that she is your "mzazi". The reason is that "mzazi" carries more weight and meaning since literally it means "the one who gave birth to you" , or "the one responsible for your birth.
kuzaa (verb) meaning "to give birth" mzazi (noun) meaning one who gives birth. Yes, even a father "gives birth" in the context that they are responsible for the outcome of a baby.
I hope the dark clouds have dispersed. :)