Continuity: When Alex walks home after writing "Paranoid Park" in his notebook, the notebook disappears (in the shot from behind) and then reappears.
Continuity: After Alex killed the man he had blood on his shirt. But from where? The blood couldn't be of the victim because Alex stood high on a train when he killed the man.
Continuity: During the scene where Alex pushes the guard who gets slashed by the trains' wheels, the guard falls backwards, so he's suppose to be lying on his back, but on the shot where Alex is standing next to him, the guard seems to be as he fell forward, and lies on his stomach.