I'm 18 years old and my boyfriend is 39. My parents are threating to have him arrested, can they do that? Is there anything else that my parents can do to stop us?

This depends on the circumstances.

If you were involved sexually with him prior to turning 18, it is possible they could file a complaint against him.

If he was your teacher, professor, or someone in a position of authority over you while you were under the age of 18 and were sexually involved with him while still a minor, they could file a complaint.

If he is someone you met online while you were a minor and he solicited you for sex, they could file a compaint.

It would be up to a state prosecutor to determine if any criminal violations took place.

Technically, at 18 years of age you are legally an adult and accountable for your own legal conduct and legal decisions. You can legally be married, joing the armed forces, sign legally binding contracts etc...