Learn French: Un flic parlait avec le chauffeur. - A cop was talking with the driver.

It is a very common informal word for a police officer, exactly like "cop" in English. It is perfectly fine for everyday conversation, but in formal situations or writing, you should use policier or agent de police.

Why is parlait used instead of a parlé?

Parlait is in the imperfect tense. It is used here because it describes an ongoing, continuous action in the past ("was talking") rather than a single, completed event. It sets a background scene. If it were a parlé (the passé composé), it would mean the cop spoke and finished, which moves a story's timeline forward.

Does chauffeur mean a private luxury driver like it does in English?

Not necessarily. While it can mean a private driver, in French, chauffeur is the everyday term for any professional driver, such as a bus driver (chauffeur de bus) or a taxi driver. For an ordinary person just driving their own car, you would instead use conducteur.

Could I say parlait au chauffeur instead of parlait avec le chauffeur?