The prince's moth is presumably deceased. His father is around, but it's indicated that there's some distance between them. (The king seems to think his son is a rebellious boy who just needs to settle down and produce some heirs already.)
Bambi': The eponymous character's mother is shot by a hunter. His father, who is distant and mysterious, subsequently begins to take care of Bambi. (This somewhat matches the life-cycle of real deer and was part of the original book.)
Mowgli's birth-parents are deceased. On top of this, the wolves that raised him appear very little. They were quite major in The Jungle Book book, according to which his biological parents were killed by the tiger Shere Khan.
In the first The Rescuers movie, Penny (the child in distress) was explicitly an orphan (though it gets better for her in the end); in the second, Cody's dad was "gone", and his mother's voice was heard twice, from offscreen. (As a matter of fact, a few fans have theorized that Cody is adopted by Marahute, the golden eagle he saves, as she's a far more interesting mother figure in the boy's life.)
Played with in The Fox and the Hound. A hunter kills a fox cub's mother at the very begining, making a woman adopt him. However it's not for long because she has to abandon him on a wildlife reserve to save his life.
Aladdin: The title lead is an orphan and homeless, and Princess Jasmine's mother is dead. We see Al's parents in one comic, and meet his dad in King of Thieves.
Aladdin had a mother in early drafts, and she even had her own song, but the idea was ultimately scrapped.
Mulan averts this with Mulan herself, but plays this straight with another character, whose mother is implied to have died prior to the events of the film, and whose father is killed offscreen.
Tarzan is an orphan; his parents were killed by a leopard.
He gets a loving mom in Kala. Though his adoptive dad Kerchak doesn't accept Tarzan as his son until he (Kerchak) is on his death bed.
Aladar from Dinosaur gets separated from his parents while he's still in his egg, and then gets taken in by a family of lemurs. They likely died in the meteor shower years later, if not sooner.
Kuzco in The Emperor's New Groove is a spoiled eighteen-year-old Emperor with no parents anywhere in sight. The only mention of this comes in this exchange between Kuzco's incredibly ancient (and recently fired) adviser Yzma and her lackey Kronk:
Yzma: Who does that ungrateful little worm think he is? Does he... have any idea of who he's dealing with? How could he do this to me? Why, I practically raised him! Kronk: Yeah, you'd think he would have turned out better. Yzma: Yeah, go figure...
Well, If Yzma's statement and implied personality meant anything, it seems we know why Kuzco was a jerk.
Atlantis: The Lost Empire: Milo's parents, according to disc 2 of the two-disc DVD of the film, were both killed in a railway accident when he was still a baby, and was raised by his grandfather, who was Preston Whitmore's best friend, who apparantly died much later (and according to Whitmore, as a broken man); while Kida's mother the Queen was killed trying to save the titular Atlantis from the tidal wave which then submerges her kingdom, and her father King Nedakh is killed by internal bleeding thanks to an injury he received from Rourke as an attempt to steal the Crystal and kidnap the princess, and in the case of the latter, it also caused her to become a Queen herself at the end of the film.
In Lilo & Stitch, Lilo is raised by her sister Nani after their parents were killed in a car accident. Stitch is also somewhat traumatized when he learns that he has no conventional family, as he was artificially created in a lab. The Mad Scientist responsible eventually becomes a father figure, though.
Treasure Planet. Jim's father Leland Hawkins outright left his family, setting sail on a ship, never to return. In fact, that big hole in his heart is part of what caused Jim to be a delinquent in the first place. Took an awe-inspiring adventure and a Space PirateParental Substitute to straighten the poor kid out.
The Princess and the Frog: While Naveen's parents are alive, they refuse to financially support him because of his partying and womanizing. Tiana's father dies in between the events of the prologue and the main film.
Finding Nemo: Coral gets eaten by a barracuda along with the rest of Nemo's siblings.
Remy's mother was written out of the script in Ratatouille; Linguini's mother is recently deceased and his Disappeared Dad is actually the famous Chef.
Up: Russell's dad is alive, but more interested in his new wife/mistress Phyllis.
Don Bluth is similarly in love with this trope. Fievel spends An American Tail trying to locate his parents, and there is a deeply depressing scene with some orphans in an alley toward the end.
Edmond from Rock-A-Doodle also has both parents but they only appear in the live action segments in the beginning and end. He asks where they are a couple of times during the animated section but never gets a response. (They're fine though.)
In The Land Before Time, Littlefoot has no apparent father, although one of the later movies introduces him. His mother, however, passes away less than halfway into the film. Cera has a father, but her mother disapears early on. Petrie only has his mother, and except in the original movie, neither does Ducky.
Spike, however, is a true case of this trope, seeing as he was apparently abandoned as an egg.
Their T. rex friend Chomper actually does have both parents, though he is separated from them for awhile at the beginning of his life due in part by Littlefoot and his friends (they were attempting to rescue an egg that the egg-eaters were trying to eat, but a mixup caused by a landslide resulted in Littlefoot and his friends unknowingly bringing a Sharptooth egg into the Great Valley).
Anastasia (or Anya) is raised in an orphanage until she's 18 years old. Here, the Parental Abandonment is fairly central to the plot and feels a lot less tacked on than usual.
And certainly, it's worth bringing up Titan A.E. in this context: the destruction of Earth effectively orphans her handful of surviving species. Humans are explicitly described as an "orphaned race". The main character is also an orphan; his mission is not only to find the Titan, but to find his father, whom he finds out about halfway through the film is dead.
A milder case can be found in The Secret of NIMH — the Brisby children do have a mother (the heroine), but their father, Jonathan, is dead.
Both played straight and inverted. Manfred is an abandoned parent - ( his family was killed by hunters); Sid's family abandoned him because he was annoying; Diego abandoned his pack; the human baby was simply "misplaced". His mom is killed by a smilodon, and he is reunited with his dad at the end.
2: Ellie was separated from her herd, and her adopted opossum mother is presumed to be deceased.
Surf's Up: Cody and Chicken Joe's fathers were eaten by a whale and fried for dinner, respectively, and Lani was apparently raised by her uncle, the Geek/ Big Z.
In The Swan Princess, Prince Derek has a mother but no father, and Princess Odette has a father but no mother. This is a particularly mind-boggling example, as Odette has just been born at the start of the movie, but there's not even a single mention of her mother — just "happily, a daughter was born," as if the stork paid an unexpected housecall.
It's probably safe to assume that she died in childbirth. But then it's odd that the moment is entirely treated as the happiest of days, when it reality there would also be sorrow over the death of the queen.
And then later in the movie, Odette's dad is killed, leaving the poor girl with no parents
Shrek: Shrek's parents were written out of the script (originally he wanted to become a knight and they disapproved); Donkey is sold by his "grandmotherly" owner (Donkey himself nearly becomes a Disappeared Dad when he takes a break from his Dragon girlfriend (who was pregnant at the time) — in his defense, she never told him (or, if she did, he certainly couldn't understand due to not speaking Dragon) she was expecting, he just knew that she was "moody"); Fiona's parents placed her in a lonely tower "for her own good"; Prince Charming has no father; Arthur's parents are presumed dead if he's Fiona's only other relative.
Inverted in Tangled where Rapunzel was kidnapped as a baby and raised by Gothel, thinking that she was her mother. Her real parents never gave up hope trying to find their missing daughter and they were happily reunited with her at the end of the movie. However, it was played straight with Flynn/Eugene, who is an orphan.
In Blade Runner, Rachel is a Replicant: a biological android without parents who has false memories of having a family implanted to give her emotional stability.
Another one of Ridley Scott's works, Prometheus, has this as its Central Theme. The "Engineers" left the human race to fend for itself, and then attempts to exterminate it. Peter Weyland built the most powerful corporation on Earth, and his daughter Vickers worked hard to be worthy of inheriting it, only for Weyland to seek immortality via the Engineers instead. Shaw discovers that she's pregnant with an alien fetus, and immediately has the autodoc cut it out of her and attempts to kill it.
Shaw: Why do you hate us?
Well, the "Engineers" are genetically identical to humans, and humanity's answer to that question seems to be; Because you're not what we wanted.
Invoked by David. "Don't we all wish for our parents to be dead?" Weyland programs him, serving as sort of adoptive father, and without him, David would consider himself to be free.
The Burns Gang of The Proposition. Arthur, Charlie, and Mikey are brothers, but there is no reference to their parents and Arthur is implied to have raised the younger two. According to an interview on the DVD, Sam Stoat, another gang member, killed his parents.
The live-action Japanese film Nobody Knows is a heart-breakingly realistic treatment of this trope: 4 children are abandoned by their feckless mother and do their best to cope in her absence.
Likewise, the French film La Fracture du Myocarde (known in English-speaking countries as Cross My Heart) deals with a boy's attempts to keep the authorities from learning that his mother has died and he's living on his own.
In MirrorMask, Valentine briefly mentions that his mother abandoned him, although "She wasn't really me mother, anyway. She bought me from a man..."
Repo! The Genetic Opera has Shilo, whose mother Marni died just after she was born. This is only a borderline case, as her father is still present and taking care of her — unfortunately Marni's death affected him too, making him desperate to hold onto Shilo and thus ridiculously overprotective.
And then there are the Largo siblings. All three of them were bailed on by their mothers (whether Rotti had them killed or they simply left is unclear) and, combined with their distant father, they're all rather screwed up. The Expanded Universe canon material found on Myspace implies that Pavi and Amber would have been okay if their mums had survived their childhoods, but Luigi was born the way he is.
Pirates of the Caribbean: As a child, Will Turner is found floating on a piece of wood. He told Jack his mother raised him, but she's dead now; his father spent a considerable amount of time tied to a cannon at the bottom of the ocean before making a deal with Davy Jones.
Elizabeth's mother apparently passed away when she was quite young as well.
Kick-Ass: The hero's mother dies of a brain aneurysm at the breakfast table.
Delayed version: In Away We Go the lead characters are forced to find another place to live when the man's "heroically self-absorbed" parents leave the country and rent their (the parents') house to strangers. Apparently their son's girlfriend being pregnant and effectively homeless wasn't a good enough reason to delay a refreshing move.
Cher in Clueless is a half orphan; her mom died due to a freak accident during a routine liposuction, although she still likes to pretend she's watching over her. Josh even ribs her about her desire to makeover Tai being a manifestation of her having no mother and treating her as a Barbie doll. Later (in one of the few played straight scenes), when she is insecure that she isn't enough of a 'do-gooder', her father tells her that he hasn't seen such good-doing since her mother.
Subverted with Josh, however; despite having divorced his mother and having no blood-relation to Josh, Cher's father makes a point of being a devoted father figure to him. ("You divorce wives, not children.")
All 3 main characters in Darkness Falls. The lead's mother was killed by the Tooth Fairy when he was ten, and since he became a ward of the state, Parental Abandonment must apply to his father too. Then later the other little kid is in the hospital, and his twenty-two-year-old sister Emma Caulfield is making the medical decisions. The parents are simply never mentioned, although under the circumstances they're probably dead.
In Doomsday, Eden's mother managed to get an army helicopter to take her out of a plague-infected Scotland when she was four years old. She had an envelope with her mother's address, so she would know where to go when she's older. A grown up Eden does eventually get to go to that address, and her mother's dead. The Disappeared Dad is never mentioned.
Two of each in Okuribito (Departures): Daigo's father had been a Disappeared Dad for so long Daigo can't remember his face, while the encoffiner's secretary reveals she abandoned her young son over a fling. Both parents were from small towns; it's implied that they still care(d) about their children but shame prevents them from returning.
Empire Records: Lucas' mother turned him over to the state when he was twelve "for being a bad seed", Berko appears to be living alone in a cabin behind the store, Gina, AJ, Eddie and Mark's parents are never mentioned, nor is Corey's mother, and when Deb is asked about her mother, she says something along the lines of "If you find her, let me know - I'd like to talk to her too".
In An Officer and a Gentleman, Zack moves in with his father after his mother commits suicide. Dad ignores his upbringing and leaves Zack to be raised above a Philippine brothel.
Patrick "Kitten" Braden from Breakfast on Pluto is the product of an illicit affair between the local Catholic Priest and his beautiful young housekeeper. As such, he is promptly given up for adoption and raised by an uncaring foster mother. The film revolves around Patrick's search for the mother who abandoned him.
The version of The Penguin in Batman Returns suffers this trope in the worst way possible. Born horribly disfigured, his parents, the rich and ritzy Cobblepots saw him as a shame to their family. As such, they took him to the park and actually threw his stroller into the river which took him into the sewer where he was found and raised by the zoo penguins. It's no wonder he hated surface-dwellers and attempted to destroy Gotham.
This tropes always thought he was actually raised by members of the freak circus.
The Japanese film The Homeless Student delivers a one-two punch: first the mother dies, then the children come home from school to find all their belongings outside the apartment. The shiftless, frigid father rides up on his bicycle to explain he has gambled away all the family's money and hasn't paid the rent. Then he rides off again, leaving his three children to fend for themselves. What.
Dom's children in Inception. Their mother, Mal, is dead, and Dom has been framed for her death and can't enter the U.S. or any country that has an extradition treaty with it. He agrees to get involved in the film's plot because it will let him see his kids again.
Sam Flynn in TRON Legacy. His mother was killed in a car accident shortly after he was born. His father disappeared without a trace. Fortunately, he did have a Parental Substitute in his father's friend Alan.
Winter's Bone has Ree looking for her father Jessup, who has abandoned is family and left them in the lurch financially to boot.
Star Trek: Amanda Grayson is a Disposable Woman who dies on Vulcan just seconds before everyone is beamed to safety, and Spock is standing two feet away and is just too late from grabbing her. Because Spock's day wasn't going horribly enough.
George Kirk dies just seconds after the birth of his son, James T. Kirk. Oh, and Winona Kirk is 'off-planet' when her young son decides to save George Kirk's old car from being sold by his creep of a step-father by driving it (and nearly himself) over a cliff into a huge deep valley - in Iowa.
In Godzilla (2014), it's implied that Joe Brody was less than a stellar parent after his wife's death. He's still in relatively good terms with his son despite this.
In The Wolfman (2010), Lawrence witnessed his mother's "suicide" as a child, and if that's not traumatic enough, his father throws him into an asylum and then ships him off to live with his aunt in America afterwards.
In Georgy Girl the protagonist deals with her roommate'sShotgun Wedding to a handsome rake, roommate's pregnancy, all before the roommate leaves her husband, Georgy, and the baby for a wealthy man; By the time the rake (who was Georgy's lover for a while) leaves them, Georgy has been caring for their baby like she was her own.