I think it makes a lot of sense to have at least one parent kick the bucket before the narrative begins - having two parents trying to constantly impose rules on the protagonist is just narratively redundant. If your story isn't about the family dynamic, then it easily gets in the way. Having one parent dead also provides a fast way to get some decent drama and have the readers feel empathy for the protagonist. You can get a lesser version by writing a divorce into the backstory, instead.
I appreciated Digimon Tamers for how much involvement and discussion it gave to parents whose children are engaging in giant battles that only they can do.
I also like Clannad for having so much focus on both Nagisa's present parents as Tomoya's dead/distant ones, especially in the after story where Tomoya realizes that he's become just like his detested father and resolves to change. That series was unusually good for representing the logistics of a high school romance and carrying it onwards into early adulthood.
I've felt that anime and anime-inspired works often just ignore the parents altogether for their high-school age casts. Since parents play a huge role in a teen's life at that age, I find that those settings feel artificial. If the parents aren't going to be important and the kids are going to spend all their time hanging out with each other, college would be a better setting, just like Ah My Goddess. That would have the added benefit of making everyone's adult-quality skills seem less absurd.
In Etherwood, I made the conscious decision to keep both parents alive and still together, which is unusual in my genre and took a good bit of work to ensure they are different characters with different narrative roles, while both being caring, reasonable parents.

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What do you think, have I succeeded?