Why is my son still having temper tantrums at age 13?

The most common... ...Cause of persistent tantrums is persistent rewards for having them. On the other hand, if his tantrums get him absolutely nothing, it may be time to see a mental health professional.
Because they work! Negative or positve attention reinforces any behavior; ignoring makes a behavior go away. You probably do need a behavioral therapist to teach you how not to engage in discussion, argument or "giving in" when he tantrums. With help, you can set expectations, consequences for inappropriate behavior & rewards for appropriate behavior. Once you start a behavioral plan, you must act consistently.