I had heard "Я не вижу ни мамУ, ни папЫ". It is not correct In Russian "мамУ" and "папЫ" simultaneously. Мay be either "...мамУ ... папУ" or "...мамЫ...ПапЫ". More correct "мамУ" and "папУ" (if "я вижу..." or "я не вижу" because my eyes is closed). But more correct "мамЫ" and "папЫ" (if it is meaned that they are not here or there and because I do not see they). I have send report.
"Both ... and" is only used in the positive sentence: "I [do] see both my mother and my father."
With "do not" or "don't" use "either... or:" "I do not see either my father or my mother." ("Do not" supplies the negative word, and the alternative negative sentence is "neither... nor," of course.)
You wouldn't say "I see either my father or my mother" unless you could see just one person, and you know it must be one of them.
Say they are both running around town in identical green, purple, and orange inflatable dragon costumes. You see one of them. Nobody else has anything like that, so it has to be either mom or dad.
It can seem that way, but it's more complicated. A verb ending in "у" is usually in first-person present tense. If a noun ends in "у", that usually tells us that it's in accusative case, which means it's the object of the sentence. It will have the same ending regardless of what kind of verb precedes it (second person, third person, past tense, etc.).
I hope I understood your question correctly.