This is another question that I am often asked. While most of those asking it are young teens, the seniors in high school preparing to go to college often tell me, "My parents insist that I get a degree so I will have something to fall back on if acting doesn't work out." These college bound seniors also have not convinced their parents that they should be actors. If they had done so they might be going to an acting conservatory or be headed for a BFA Acting program at a good university.
The problem is long standing. It is almost traditional for parents not to want their children to become actors. So many actors get bad publicity, are involved in affairs, substance abuse and divorces that the profession gets a bad name. Historically, actors have been looked down on and considered no better than thieves and prostitutes by the government, the church and the public. This is so even when an actor like Ronald Reagan became President. Puritanism still has a strong influence on the public when it comes to shielding offspring from harm.
The answer that I am going to provide for the problem will not be popular with all of my readers. The way to convince your parents that you should take acting lessons or go to acting school or to become an actor is for you to have demonstrated to them that you truly have the outstanding talent and personality to succeed at acting. If your participation in school theatre, competitive acting events in the school speech program, and other drama events at school and in the community do not rise to the level of persuading your parents that you have sufficient talent to succeed, I am afraid that there is not much hope for you to ever convince them. If your parents do not believe in you, then who will?
I have in the past recommended that young people seeking to convince their parents about their participating in acting should read all they can about the profession so they can answer their parents' questions when discussing it. But this is just discourse. To be convincing, you must show your parents that you have what it takes.
I really hate to be less than encouraging, but acting is a tough life. It is tougher than can be imagined. Until one tries it, he or she can only say they know that, but they will not know the reality of it. However, I must be even more discouraging to the aspiring actor. It is truly rare that someone comes along who has the look, the courage, the charm, and the talent to succeed. It is easy to become enthralled with the idea of being an actor, but it is a long way from the chorus in the grade school musical to a professional career. The fact is that most who aspire to become actors do not have what it takes.
The reason why most parents need convincing is that their children are just not talented enough to succeed. The parents can see that and they insist their children to to college where they can learn to do something at which they can succeed. It is a pity, but real talent which includes personality and look does not exist is most people. I will not be so popular as I answer this question in the forums in which I participate. My answer will be, "If your parents need convincing, you probably do not have enough talent to succeed."