The villain responsible for the deaths of Thomas and Martha Wayne is Joe Chill.
In the best-known Batman origin, Joe Chill is a small-time criminal who attempts to rob the Wayne family in Crime Alley. He shoots Thomas Wayne after the doctor resists, then kills Martha Wayne when she cries out. Their son Bruce survives the attack, and the trauma leads him to dedicate his life to fighting crime as Batman.
The murder is the defining event of Batman's origin, but the story has changed across comic-book eras and adaptations. The original 1939 account showed an unnamed mugger; the killer was identified as Joe Chill in Batman #47, published in 1948. Some films and alternate continuities use a different killer or leave the identity unknown, but Joe Chill remains the standard canonical answer in the mainstream Batman mythos.
The Joker is sometimes associated with Bruce Wayne's tragedy because he is Batman's most famous enemy, but he is not generally the person who murdered Bruce's parents. Ra's al Ghul and Harvey Dent are likewise important Batman characters, yet neither is the conventional answer to this origin question.