I'd say that the main reasons of absentee parents in anime come down to something pretty simple:
1) If the show is focusing on characters old enough to be independent, then it's generally assumed that they aren't living with their parents anymore (and thus the parents can be no more than side-characters).
2) If the show is focusing on characters too young to be independent, then it is rather likely to be a "school focused" show, (since that is where the characters will probably be for most of their time and interactions). The parents in most cases aren't going to be at the school, thus preventing them from being much more than side characters.
Either way you end up with parents that aren't really relevant to the story as a whole. This means that oftentimes it just comes down to a gamble of what exactly gets shown in the very slim amount of time that the main characters are home, or whether or not we get a "tragic parent accident" arc happen in the case of independent characters. Heck, even characters we've already mentioned often fit in this way. For example in Puella Magi Madoka Magica, Madoka's mother is on screen for maybe 15 minutes out of the entire 12-episode season, and like 90% of that is her either answering Madoka's questions or worrying about her, and she's one of more concrete examples.
Thus really the only times you are going to get a fair bit of parent characterization is when a show does one of about four things that can bring parents and children in contact with one another as individual characters rather than just "protagonist" and "protagonist's parent". (I was going to put some show examples here, but a fair number of them were significant spoilers so I deferred.
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1) The kids became independent on their own, and then go into a business that one or both of their parents were involved in/keeping secrets about.
2) One or both of their parents are immortal/long lived. This often has crossovers with #1.
3) It's a generations show, especially in a sequel style. If you've got a show about the "next generation" the parents are usually already pretty well characterized, which makes it easier to write more using the same details.
4) It's a show about characters right at the verge of independence, and their is a parent/child desires conflict arc.