I already see a few naysayers here. ...[cracks knuckles]
Wrongo!!! TMNT 1989 is ALL about fatherhood. There is a right answer, and you guys got it wrong. lol Chris, you lost because you only focused on one particular relationship (Splinter and his turtles), when almost every character in the movie applies.
The Shredder: is creating a gang by enlisting neglected children and teens by offering them power and freedom their parents wouldn't. Specifically says "This is your family. I am your father." He says it in the exact same intonation as Darth Vader in ESB, which some might call a rip-off, but what I think was an intentional homage for the sake of contrast (especially considering the pre-existing similar character design). When Vader says it, he's being honest and literal, when Shredder says it, he's being disingenuous and manipulative.
Tatsu and The Foot: aforementioned teens who've turned to crime to partake in the freedoms enjoyed at the warehouse. They've all rejected the establishment and confines their parents imposed. No rules. "GO. PLAY." Tatsu is also an example of the evil manipulation afoot (nudge, nudge). In the training room, after he kicks that kid in the face, he says, "never lower your eyes to an enemy" making it very explicit that these men, in the guise of family, are exploitive and dangerous. Later, he returns to the same room to throw a tantrum and kicks the shit out of a bunch of the teens, knocking one out (killing him in the original script), and giving the kids a chance to mull over the idea that this might not be the family they're looking for. Tatsu's character can easily be drawn as allegorical to abusive fathers who's simple-mindedness and lack of temper control can destroy the idealized family they've worked so hard to build, on top of blatantly hurting the younger and more innocent.
April: lives in an apartment above a shop that her father used to own. Keeps running the business even though it's losing money because she misses him. Has to deal with the loss of all of it when it burns down, paralleling the loss the turtles are experiencing. They then retreat to the farmhouse she grew up in. That's probably her father's truck Casey and Don repair.
Casey: a (possibly homeless) man-child who played games for a living until injured, after which he became an aggressive vigilante, lacking sophistication and using unregulated violence to combat the ills he perceives of the world - representation of society's fears about boys who grow up without father figures.
Splinter: his whole motivation for raising the turtles as ninjas was to restore honor to his family in lieu of the murder of his father figure (former owner), Hamato Yoshi, who was killed by Oroku Saki (The Shredder). When the turtles fight the Foot, we're witnessing a war waged by fathers (over the death of one of their fathers), but fought by their children.
Danny: a non-canon character with his own feature-spanning subplot? What could the significance of that be? Oh wait, it's because he runs away from his father(April's boss, who is also the one who fires her as her father's shop is burning) when his father faces a fathering dilemma and shows flawed effectiveness and befuddlement at how to deal with the situation, spurring Danny into the middle of a conflict between the Shredder and Splinter (each idealized and opposing father figures) who he spends the film switching his allegiances to and learning from, in the end reuniting with his own father and forgiving him his flaws as well as acknowledging his own. Both Danny and his father then correct wrongs they've both incurred on April.
I don't even need to get into the turtles themselves, because their personalities and individual relationships to Splinter are all well known and on display throughout the movie (well mostly Raph and Leo, which is probably why they included that chat between Don and Mike when they're waiting for pizza at the beginning, so they could just carry out comic relief duties for the rest of the film).
The Last Crusade had Dr.Jones Sr. and Jr. bickering cutely. And skronking the same chick. I bet they high-fived after the credits.