In My Immortal, Tara's parents find her so annoying that they leave her in the middle of the desert.
Popular in earlier Code Lyoko works and Epileptic Trees in some form or another due to the sheer number of unseen or unmentioned parents. Now that all of the parents (that are minor characters,anyway) have been proven to be alive and well ((except for Sissi's mother, who is never seen or mentioned) this has fallen into disuse.
A Crown Of Stars: Shinji and Asuka's fathers dumped them by the side of the road shortly after their mothers' deaths before they were four. Fifteen years later they are still traumatized about it. In chapter 33 Daniel and Rayana talk about their fathers' behaviour, concluding Asuka's was the worst of the lot.
“Considering he may have been the worst dad out of your whole group, we’ve left your father alone for now, Asuka,” Daniel said. “Let us know what you want done later.” “And when the group includes Ikari Gendo, that’s saying something,” Rayana said. “Gendo was at least a bad man trying to do good, just really badly. Pieter Langley had no such excuse. I mean, cheating on your mom in the next room with the doctor? Being married to her by your mother’s funeral? Abandoning you after that? Say the word, and I’ll smite the piss out of him.” ‘Oh yes, my wonderful and supportive father, who ditched Mama before she was even gone, and me before she was two weeks in the ground. He can rot in Hell in the cell next to Shinji’s father. They can swap ‘cheating on your child’s mother’ tips.’ “He can stay wherever the Hell he is now. He’ll get all the care from me that he gave my mother and I,” she said rather bitterly.
Advice And Trust: After their First Kiss Shinji blurts out accidentally he was abandoned by his father when he was a little kid after his mother's death. Amazed, Asuka muses that is exactly what happened to her. Both teenagers talk about how they never got an apology or explanation from their fathers, how they still miss their mothers and have nightmares about their deaths... and start bonding thanks to their shared griefs.
The Child of Love: Shinji and Asuka were abandoned by their fathers shortly after their mothers’ deaths. After Teri’s birth Asuka is willing to try some sort of reconciliation with her family, but Shinji no longer acknowledges his father.
Shinji (in an exceptional moment of openness):"When I was four, I was abandoned by my father. Not that I ever saw him that often even when Mother was alive...but...I can still feel the pain, Asuka. The loneliness...being rejected like that. I don't want my daughter to ever feel the same way. Not if I can make sure it doesn't happen. If you'll allow me to be there..."
In Children of an Elder God, Gendo abandoned his son Shinji after his wife's death and only summoned him back when he needed him to kill Cthulhu and co. Shinji wants to believe that his father had a good reason, but Asuka thinks Gendo is just a bastard.
Evangelion 303: Shinji's mother died when he was a little child, and shortly after her death he got estranged from his father. By the start of the doujin, he has not ever seen Gendo in ten years. Nothing is known about the parents of the other pilots, but given Asuka's reaction to Shinji's history about his parents and what happened to her in the original canon, it is likely that she is also an orphan and/or does not get along with her parents.
Jessika is buried together with her father, so she had also lost at least one of her parents.
HERZ: Most of Shinji and Asuka’s traumas and psychological issues come directly or indirectly from having been abandoned by their fathers when they were four. Shinji has forgiven his… but Asuka’s relationship with hers is non-existent.
Last Child of Krypton: Shinji and Asuka’s fathers abandoned them shortly after their mothers’ deaths when they were four. After ten years Shinji was summoned by his father and he thought Gendo wanted to see him, but when he failed on synching with an Eva Gendo intended to send him back without even saying one direct word to his son. In the rewrite, when Shinji watched Kal-El's recording, Kal-El told he was his genetic donor and hoped his biological father raised him properly. Shinji snorted.
In Neon Genesis Evangelion: Genocide, Shinji and Asuka were abandoned by their fathers when they were three. And that was the beginning of their many psychological issues.
The One I Love Is: In chapter 9, as Asuka is going through a terrible break-down, Shinji finds out her mother died and her father does not care about her... and he realizes she is just like him.
Scar Tissue: In this story, Shinji and Asuka are very, very, very badly damaged. Both lost their mothers when they were barely four, and their fathers abandoned them shortly after. And that was only the beginning from their hardships. When Shinji is not blaming himself for everything he blames his parents and declares he wants nothing to do with any of them. Asuka, on the other hand... her father flatly refuses to look after her or take responsibility for her deteriorated mental state.
Superwomen of Eva 2: Lone Heir of Krypton: Asuka's father never loved her and abandoned her right after her mother's death. Her mother's doctor theorizes the man was resentful of not being her biological father since Asuka she was conceived by donor sperm. After arriving to NERV Asuka also was abandoned by his father.
Elizabeth Quatermain's mother died at her birth, and her Great White Hunter father didn't think he had the chops to raise a baby girl on his own, so she grew up in the care of her mother's spinster sister. She wasn't isolated from her father and older half-brother, but it was mostly limited to holidays.
It doesn't happen outright in The Odyssey, but Ulyssian has been noticeably ostracized by House Sesus due to his failure to Exalt.
In 'Nexus' Jack is understandably pissed when his mother tells him that his father left them for power.
In the first story of the Daria fanfic series, John Lane, Helen learns of John's parents' Hands-Off Parenting style, which means they have been away from home and their children for months, and has no choice as an Officer of the Court but to call the police on them for child abandonment.
In the Ah! My Goddess fic Ah! Archfall! Lind was abandoned by her mother for having a single wing. It was actually because her mother was forced to abandon her, on pain of death after it was discovered that she was the Almighty's bastard daughter. The Almighty himself is forbidden from telling her due, on literal pain of death.
After Naruto's father was confirmed to be the Fourth Hokage, fans speculated on what had happened to Naruto's mother. Naturally, fanfics featuring Naruto's mother abandoning him sprung up. To make things worse, there are fics in which the Uzumaki clan survives, and the clan as a whole, including Naruto's mother, abandons him. Two such examples of the latter can be found here and here.
Used in the Pokemon fic When Ash Was A Kid, a prequel to one of the author's other stories. It depicts a young Ash's father being too busy to see his own child, and thus Ash acts out and grows alienated, soon even being put on medication in an easy to miss paragraph.
In Empath: The Luckiest Smurf, Empath seems to think this about Papa Smurf when he reveals the true nature of their relationship in the novel. Empath is Papa Smurf's only begotten son, but was brought to Psychelia in order to be trained in the use of his abilities.
Ran has been taken care of by Yukari since she was a baby after she was abandoned on Yukari's residence.
In Gensokyo 20XX, later in the series, we meet Maribel and Renko, two very small children, and the natural question is: What the hell happened to their parents? Well, we never find out and being that the story in which meet them in this particular series, Gensokyo 20XXIV, takes place in the aftermath of a nuclear war, it is a safe assumption that they died when the bomb was dropped or of radiation sickness, though, Yukari makes note that both of the two were fairly healthy and well-fed, suggesting that someone took care of them but died or the two were close to a food source, the which Maribel and Renko confirm in their recollections on the note of the food source.
In Foundling (a branch-off), we have Reimu again and that is because she, for some reason, wasn't wanted, leaving her to be taken in youkai. Apparently, she wasn't the first human child to be abandoned in that particular forest, as the practice of doing so is age-old. However, it is implied that she might've been the first one to have survived. This is also the backstory of Ran,Chen, and Yukari like their previous incarnations.
Chico, a supporting character in Yu-Gi-Oh: The Thousand Year Door, Redux was a victim of this, and was old enough to remember his parents doing this; he nearly starved after being left by his parents in the New Mexico desert, but was rescued by a Native American man who later adopted him; the police were never able to find his parents, and he never even found out why. This uncertainty about why he was abandoned makes him bitter and anti-social.
In "Dear Scootaloo", it is mentioned that some pegasi (much like the Spartans) abandon their foals for appearing too weak or sickly. This seems to have happened to Fluttershy, Derpy, Rainbow Dash and Scootaloo. Dash always preferred to think her parents had died and made sure she would be left in good hooves, making it very difficult at first to accept the evidence that Scootaloo is her sister.
In the Longingverse fanfic "Halcyon Hearts" Pipsqueak reveals that his parents abandoned him on the Summer Sun Celebration. He heard them talking earlier and apparently his dad always told his mom she should have gotten something that started with an "A" and couldn't think of what it could mean besides 'adopted'. Fluttershy ends up adopting him herself when she finds out.
In the Criminal Minds fanfic To live again, Annie grew up in an orphanage with no information about her real parents (it's revealed later that she was taken from her mentally ill mother who couldn't even take care of herself, much less a baby and ended up in an asylum, but her father remains unknown, so most likely she was born from an one-night-stand). This trope occurs again with her sort-of foster "family" abandoning her one by one, and especially with her "mom", Strauss (which is a messy story). No wonder she has issues.
Satsuki, Nui, and Ryuuko's parental situation in Cellar Secrets didn't initially start out this way. At first, it was just a Disappeared Dad situation, with their father, Soichiro (his name isn't mentioned in the fic), being mentioned as to have left and never returned but, as time went on, their mother, Ragyou lost her mind and was interned in a mental hospital and, shortly after, her secretary, Rei, commits suicide, leaving Satsuki and Nui to find out that their father had passed away, sometime after he left and their mother dies roughly two and half ears since her interment. Fortunately, someone their father knows, Aikurou, steps into help.
...the Boarding School, where pupils will be housed, provisioned and looked after by the Guild 24/8 during term time. For those who express a need, are from overseas, or who are acting out of a purely human parental need to want to see less of their children, we can board pupils outside recognised term-times. Naturally this attracts a premium on top of the standard Boarding fees.
In Thousand Shinji, Shinji and Asuka were abandoned by their fathers when they were kids. Asuka stopped acknowledging her father's existence (even though she secretly wishes that their parents were back to protect her), and Shinji plotted how paying his father back.
The Second Try: Gendo abandoned Shinji when he was four. As far as Asuka –who was also abandoned by her father right after her mother’s funeral-, is concerned, Gendo has no right to talk about his granddaughter after abandoning Shinji. Later on, Shinji stated that his father never understood that you can love another people only because they’re related to you.
Once More with Feeling: Shinji and Asuka were abandoned by their fathers. Shinji hates his for it, and Asuka doesn’t want anything to do with hers. After a while, Shinji decides that Misato, Asuka, Rei… are his real family and cuts ties with his blood relatives.
Shinji: Three thousand, five hundred and forty nine. It's the number of days since my Father sent me off to live with my uncle. It also happens to be the number of days since anyone from the rest of my extended family bothered to so much as acknowledge that I even existed.
In the Undertale (Fan Novelization), Frisk is left alone by Toriel with instructions to stay put until she returns. What follows is a memory of Frisk's mother saying the same thing before leaving her child behind and comments that suggest Frisk was living in the streets. These memories prompt Frisk to disobey and go seeking out Toriel.