Occurs in The Lion King Adventures. 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.
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 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 20 XX, 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.
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.