Regarding “would you call [Dad or Mum] a name?”, I think a term like cognomen or sobriquet is more appropriate than appellation or name.
• cognomen, “a nickname or epithet by which someone is identified; a byname; a moniker or sobriquet”
• sobriquet, “A familiar name for a person (typically a shortened version of a person’s given name)”
Note, while epithet has a well-known sense (“An abusive or contemptuous word or phrase”) with negative connotation, that is not the sense used in the definition above of cognomen. Instead, the relevant sense is “A term used to characterize a person or thing” or “A term used as a descriptive substitute for the name or title of a person” (in which sense epithet also can serve as an answer to the original question).