Student or Learner
I don't like neither of my parents. = I like both my parents.
Do they convey the same meaning?
Your double negative to create a positive is in a form we don't use.
I don't dislike either = I dislike neither = I do like both.
While we do say "I dislike neither" we don't say "I don't like neither."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.