"I do not see neither mom nor dad" --> You are correct in saying there is redundancy by using both "do not" and "neither" because you have then created a double negative; so it is both unnatural and could convey the opposite meaning. "I see neither mom nor dad" --> This sentence is grammatically perfect but I would say that "nor" is quite old fashioned nowadays. "I can't/don't see mom or dad" --> This would be the most natural and simple way to express this.

No, the first one is odd. If you use 'not', you don't use the 'neither/nor' construction. The first one should be "I do not see either mum or dad'. To be honest, 'neither/nor' is getting a bit archaic. People don't really use them, and particularly not in an informal situation, such as talking about your parents using informal names.