|“||We will never know when she might begin to kill, am I right?||”|
–Yuka's Mother, in a sad moment of misguided concern about Nana
Yuka's Parents play a pivotal, if minor, role in the Elfen Lied series. While they're rarely seen, moments involving them provide important elements to the backstory. In particular, it is Yuka's mother who holds the most importance between the two of them, as she is the one who tells Yuka that Kouta may stay at the Maple House, the main setting of the story, for free so long as he keeps up with the housework.
Mentioned before, Yuka's parents show up very seldom, but these appearances are also indicative both of the fictional world of the series itself and of the challenges of creating any fictional series. Like many secondary characters in this series, their names are never revealed, either family or given. In the original Japanese manga, a form of casual address between Kouta's father and Yuka's mother offers some indication that he is in fact her elder sibling, though no direct dialogue ever confirms this.
Like Kouta's mother, no mention is ever made of Yuka's father. He is absent from the manga entirely, and his existence at all is only implied. It isn't known which parent holds the title to Maple House, or if they owned properties besides that one, but Yuka's mother, at least, appears to have the final say about its affairs since she is referred to as its landlady and is the one to offer it to Kouta (through Yuka) in the first place. After Nousou's assault on the Maple House, Yuka's mother also mentions having the residents stay with her at her house.
While Yuka's mother does not appear onscreen in the anime, being only mentioned, there is a possible animation mistake that shows a man who might be Yuka's father. During Yuka's flashback in the taxi near the end of Episode 2, a still scene is shown of Kouta, Kanae and their father standing at Gokurakuji Train Station before they leave Kamakura. This scene is inconsistent with its later showing in Episode 12, namely in that there is a man in a yukata behind Yuka, presumably her father, in Episode 2, but this man is absent when the scene is further expanded upon in Episode 12. In the manga's version of Episode 12's flashback, Yuka is accompanied by her mother; in the anime, she is alone, with neither her mother nor her "father" in sight. Since Episode 2 was an early episode, certain plot details and character models were likely not yet nailed down. It is possible that they meant to show Yuka's father, which this man must have been due to his proximity to Yuka, but then either realized or were told by Lynn Okamoto that the character never makes an appearance in-series. The fact that it was also merely a still frame is another indication that it was made in haste. Due to the figure's masculine appearance that doesn't fit with the manga's showing of Yuka's mother, it clearly could not have been her.
Yuka's Mother appears briefly, if memorably, in the manga on very few occasions. She is directly referred to as the landlady of Maple House, offering it to Kouta on the basis of a maintenance agreement. Presumably, she and Yuka's father are Kouta's aunt and uncle on his father's side, but while Yuka at least once calls Kouta's father "Uncle," Kouta never interacts with Yuka's mother in either version, and only twice is even in the same room with her. The first time is shown in Yuka's mother's living room during Lucy's flashback to meeting Kouta when they were children, though this scene would, narrative-wise, be outside the scope of Lucy's memories. The second time is while Kouta is in the hospital recovering from the invasion of Maple House late in the series, and he cannot interact with her since he's unconscious. In at least one instance, Kouta calls a phone number asking for Yuka, only to be told she is not there. He never says or indicates whom he is speaking to, but if Yuka is expected to be there, it seems likely he was calling her mother's home and speaking to her mother.
One possibility is that the amnesiac Kouta has as much trouble remembering Yuka's mother as he does anything else from that part of his life, though since he remembered Yuka and seems to remember other events prior to that summer, it is unlikely he would forget his aunt. Another possibility is that in many places, especially in Japan, "Aunt" or "Auntie" can be seen as an age-related pejorative, calling a fully grown woman as being old and unattractive, and even if they're related, some women would insist they not be called as such. The only certainty here is that Yuka's mother offered to help her daughter's cousin, the son of her sibling/sibling-in-law, with an offer of a place to stay.
Still in the manga, besides the living room scene from the fateful summer Kouta met Lucy, Yuka's mother is seen holding Yuka's hand at the train station as Kouta, Kanae, and their father depart Kamakura to return home to Hokkaido. If Yuka's mother was a sibling or sibling-in-law to Kouta's father, it's highly likely she was asked to identify the bodies of Kanae and their father after young Lucy boarded the train and murdered them. It is also possible she made arrangements to have Kouta hospitalized in a mental care facility unless he was instead taken care of by other relatives. Since she is Kouta's aunt, she, by all accounts, should have been able to take care of him unless his unmentioned mother or other, closer relatives objected, but considering Kamakura was the site of his trauma, it's likely he was not kept there for fear of agitating his unstable condition after the murders.
Yuka's mother is last seen in-series directly after the invasion of Maple House, where she makes it clear she's not comfortable with Nana staying with the group, as she fears Nana would reveal herself to be just as much of a threat as the newly-recaptured Lucy. In this scene, the mother seems to be familiar with all the residents of the house at that point besides her daughter, including Kouta and Nozomi. This implies that she was asked for her approval of each new resident, but her level of contact with any of them seems in question, since Yuka is the only one she addresses, and she speaks of both Lucy and Nana in a distant manner. Whether she knew of the two Diclonius girls' horns is uncertain, as is whether or not she or anyone besides Kouta knew Lucy/Nyu was the one responsible for the murders of Kanae and Kouta's Father at this point. As a landlady, she is understandably upset by the destruction of her property and the trauma caused to her family, and sees Nana as a liability in that regard. While Yuka argues loudly on Nana's behalf, this subject is never followed up on later, and its outcome can only be broadly inferred from the series' conclusion. Nana is present to help Yuka, Mayu, and Nozomi repair Maple House, but after Kouta returns home, Nana is later seen at the grave of Mariko and states she and Kurama are now living together. It's never made clear if Kurama was allowed to live in Maple House or if he and Nana move to a separate dwelling. In addition, Nana is seen with all the others from Maple House at a summer festival sometime following Lucy's death.
If in fact Yuka's Mother was concerned about the possible danger Nana presented, it would seem odd to have her so nearby and still an effective part of the household, particularly if, as Kurama indicates, he will be away till matters concerning the Kakuzawa family's plans can be attended to. Possibly he intervened with Yuka's mother, or, the Japanese government, wishing to keep these matters quiet, arranged with the mother to keep Nana where they know she can be found, in exchange for large scale clean-up of the damage done to her property at Maple House. Some sort of arrangement seems likely, since, despite their 'harboring' of two Diclonii, one a known fugitive, no resident of Maple House was described as facing charges. Also possible, and simpler to assume, is the idea that Yuka's mother relented to her daughter's judgement of Nana's character and allowed her to stay.
Like with many characters, whether they're main, secondary, or minor characters, no final fate is ever described or mentioned for Yuka's mother. Another possibility for the Nana eviction subplot not being pursued is that Yuka's mother was killed by Lucy's dying attack on Kamakura and the world. It is not known if she lived long enough to meet her presumed granddaughter, Nyuu Jr., and any conflict with her daughter for naming the child after a friend who brought so much chaos to their lives is also not known.
On a real-world basis, it seems most likely that the Nana eviction subplot was not pursued due to the demands of the manga's concluding storylines, as well as the awkwardness of the group coming into conflict with a character largely invisible for most of the series.
Just as the protectiveness of Kouta's father showed up in his son's personality, so too can it be said that Yuka takes after her mother, possessing a stubborn determination.