there is a pet dog that is mentioned throughout the series and hangs around a main character. after the apocalyptic event the dog is no longer mentioned, but it is not implied or mentioned that it died.
There aren't depictions of self-harming behaviors in the story, however, in one of the songs, a character references that he and another would burn themselves with cigarettes.
Not a parent, but an older sibling:
Over a decade prior to the events of the series, a girl (preteen - early teenage) goes missing when left in the care of her brother and brother's friend. Instead of ensuring that she makes it home safely, the boys went to a party. This event is referenced multiple times throughout the series, and is a sort of catalyst for parts of the story.
One character is left unconscious after a car accident. Later, another falls unconscious after quickly standing up and hitting his head on a ceiling beam
spoilers: somewhat -- humans are turned into bug creatures (that still resemble the human form) and they are heavily implied to want to eat the human survivors. a character gets his stomach bitten open by one of these creatures in an attempt to be eaten.
direct stalking occurs in one instance as a character is followed to his home. a lot of the plot revolves around creatures hiding and "hunting", and can be viewed as stalking also.
Drug usage is frequently mentioned, and occurs regularly during the events of the episodes. This is most often marijuana, however hallucinogens and morphine are also used.
Body horror is one of the main themes of the series, first appearing in episode 1 and growing increasingly prevalent throughout the rest of the episodes.
the motivating factor for one character to return home is for his parents' funeral.
SPOILER: his parents turn out to be alive, but he leaves them behind and it's heavily implied they died afterward.
While there are no scenes in which characters have sex, multiple (somewhat successful) attempts at seduction are made. This is to the extent that a nearby character is convinced sexual actions are occuring. Additionally, a character is noticably masturbating at one point, and language of sexual nature is used often throughout the series.