It depends on the expectation. If I expect that you have zero or one, you would use the singular. If I expect that you have zero, one or more, you would use the plural.
I have no father. I have a father.
I have no mother. I have a mother.
I have no parents. I have one parent. I have two parents.
The normal expectation is to have one father, one mother, two parents. So you use "no" with singular for father or mother, and with plural for parents, and "a father", "a mother" vs. "one parent".
There may be two people you call "father", for example your mother's ex-husband and her current husband, or two gay parents. As long as that isn't expected by the person asking you, you would give the same answer, but of course the answer could be "two fathers". If I asked "was your father at the football game" and you have two people you call "father", you might answer "One of my fathers was at the game."
A situation where I expect several: "How many fathers and mothers were at the school meeting?" might have an answer "There were no fathers but seven mothers present". "No fathers" has the plural because it was likely that there were several.