Is it permissible to kill a person who caused the death of one's father by running him over with a car, especially since the incident was the result of a race and he has not yet been justly held accountable?

You are not permitted to kill the driver of the car. If his killing of your father was accidental, then he must offer expiation (kaffarah) and pay blood money (diyah). If it was intentional, then the government is obligated to execute him by way of retribution (qisas), and you do not have the right to carry out the retribution yourselves. As for the driver abandoning your father and not assisting him, that is a grave error, but it does not make him an intentional killer if the accident was accidental to begin with. You are not permitted to kill him for not assisting your father.