Basic Trope: The parents are never referred to by their real names.
Straight: The parents are called "Mum"/"Mom" and "Dad".
Exaggerated: The kids are only called "Brother," "Sister," "Son," "Daughter," and similarly with uncles, aunts, and cousins.
Because the child's last name is known, the last name of the parents is known. Last Name Basis is used, even when First Name Basis would have been more natural.
Only one of the parents is named.
Justified: The protagonist is so young that he has yet to learn that his parents have names other than "Mommy" and "Daddy." See Most Writers Are Adults.
Unlike the conventional practice in American culture and elsewhere, the parents want their kids to call them by their names just like they would for anyone else.
The parents' names are known to the audience, but their son and daughter are only addressed with pet names ("sweetie," etc.) that don't even qualify as nicknames, and are only referred to as "Alice and Bob's son/daughter."
Subverted: The show uses several obvious devices to avoid saying the parents' names, then goes ahead and reveals them.
Double Subversion: But the supposedly revealed names are immediately revealed to be wrong ("Why are you always forgetting my name? It's not Alice!") and the audience still doesn't know their real names.
Parodied: The parents' names are actually "Mom" and "Dad."
The parents are left nameless, unintentionally, just because they aren't that important. At a later stage of the show's history, they are more important, but still don't have names, and the writers resist giving them names because it would be awkward. Finally, the writers go ahead and decide on names for the parents and reveal them to the audience.
After the parents are named, they furtively look at the calendar. When the big day comes, they hold a party. In is 7 years since they committed that crime. Statute of Limitations. They can revert to their "original" names.
Averted: The audience knows the parents' names.
Enforced: "The parents really aren't that important to the show. It really works better if we don't give them names."
Lampshaded: "Hey, come to think of it, I don't actually know my dad's real name."
Invoked: The character is the kind of preschool-aged kid mentioned above as an example of a Justified Trope. He is kidnapped by someone intending to hold him for ransom, but the burglar finds that he can't simply ask the kid who his parents are, because he doesn't know.
Defied: Whether because he's very young or because he's The Ditz, the character turns out not to know his parents' names. Someone who does know immediately corrects this gap in his knowledge.
Conversed: "I have a life of my own, you know. I'm not like those mothers on our kids' cartoons, who don't even have names of their own beyond "Mommy."
Deconstructed: A culture with strong Planet of Steves tendencies in which all mothers and fathers have their names changed to "Mother" or "Father." What were their names before they were "Mother" and "Father," you ask? "Husband" and "Wife," of course.
Reconstructed: But in such a culture, the kids themselves still use more familiar names. For instance, the father's name is "Father," but his kids don't call him "Father," they call him "Daddy."