His protagonists are people who aspire to make decisions and believe they have free will, but whose lives were crafted and predicted far in advance by forces way beyond their control. And so they are all destined for tragedy even as they consciously take actions to keep their lives in check.
For example, Joint Security Area is about North and South Korean soldiers who cross the border between their nations and become friends in secret. Their friendship is a self-aware act of rebellion against cultural and political restraints, but even as they "independently" act against this system, their inherent training and the pressures of reality have been internalized to a point where it is too late for them.
In the end, it isn't even that their commanding officers break apart their secret meetings. Their friendship implodes because they cannot change the simple fact that they are driven to fight each other, and so the story ends with mutual destruction. Contrast this with something like Welcome to Dongmakgol, which also features a friendship between North and South Korean soldiers, but which presents the rivalry between nations as a divide conquerable by reason, humanity, and optimism.
Oldboy drives the point home further by having the protagonist literally be the product of the antagonist's creation. He is a guy who is locked in a hotel room for years and force-fed media, drugs, and hypnosis to physically and mentally turn him into exactly what the antagonist wants him to be. When Oh Dae-su finally gets to leave his prison, he thinks he is free and that he can live his life again, but there is not a single moment in the movie when he isn't playing out his life exactly as the antagonist had intended.
Park Chan-wook must subscribe to the theory that consciousness is an illusion; that we are all a collection of inter-locking mechanisms and we only think that we think, but in reality we are just the consequences of our programming and our environments.
It's not an altogether bad thing. In fact, it can be kind of liberating to realize that certain choices are simply not available to you. This realization can remove quite a bit of uncertainty - and on top of that, you realize that you cannot "fail" at certain things if you could never get started on them in the first place.
Which brings me back to Stoker. It's probably the only Park Chan-wook movie I've seen so far with a happy ending.
This time around, the protagonist is a young woman, India, who just turned eighteen. India's father has died in a car accident and her uncle Charlie has come to live at her house. While dealing with the changes in her family, India is also coming to realize that Charlie is a weirdo and - surprise - a murderer.
At one point, India is attacked by a classmate and nearly raped. Charlie intervenes and kills the classmate. Later, India masturbates to the memory of her classmate's death, and this is around the point where she starts to realize that she possibly also has some psychological problems.
India keeps digging into Charlie's background until she learns that not only is Charlie responsible for murdering multiple people in the last few weeks, but he also killed her father and staged it as a car accident. On top of that, he has been sending India love letters for the last eighteen years and planning for the day when he could escort her away to a new life as his lover. Virtually all of the events of the preceding few weeks have been part of a devious scheme by Charlie to bring India into his life as his murder bride.
This is around when the movie seems like it might be going the Oldboy route in which Stoker, having learned that she is a pawn in the antagonist's long-awaited plan, resigns. But instead, she accepts her fate in a slightly different way: she realizes that she has always been doomed to be a murderer, and she chooses to accept this on her own terms. So she kills Charlie and skips town in order to - I assume - kill whoever the hell else she feels like, starting with a sheriff who pulls her over for speeding.
I guess it's not really a happy ending, but it's as much free will as she could really hope for, given the circumstances.
(Side ramble: I wonder if it ended like this because of an American influence? Americans really love the idea of choice, even if we rarely have it, so I can see a studio exec making a last minute change so that Stoker isn't completely pre-destined.)
Ultimately, I liked Stoker, and it's definitely got Park's style all over it. It feels kind of like what Natural Born Killers wished it could've been. Even though it feels a bit tedious at times, it's definitely worth your time.