The most important thing a parent can do to support their scholar's arts education is to show that they genuinely value and appreciate it.
Parents understandably want their children to get a good education that prepares them to support themselves in the world when they come of age. Sometimes, this unfortunately causes parents to devalue the arts (what good will this do them in getting a good job?) and either directly or implicitly discourage them from participating in it. This, again, comes from a low level of sophistication of aesthetic understanding. There is often the case that the parents only experience the arts in their lives as a sort of cheap entertainment, a functional distraction from the "real" efforts and problems in life. And why would any good parent encourage their child to spend a lot of time and effort on distracting themselves in school? Even supportive parents sometimes betray this belief with comments like "I just hope she is having fun in choir, that is all I care about," or "I don't understand it at all, no one else in our family has been into art, but he likes it so we support him in it."
Parents, please, always remember that many of the most successful people in history (Thomas Jefferson, Thomas Edison, Albert Einstein, Stephen Hawking) were artists and musicians as well as hugely successful scientists, public figures, and entrepreneurs. Developing determination and aesthetic sensibilities in the arts does not handicap a person going into another field like medicine, law, or business, but rather greatly improves their character, creativity, and drive to be successful. The famous violin teacher (and friend of Einstein) Shinichi Suzuki said "The purpose of [music] education is to train children, not to be professional musicians but to be fine musicians and to show high ability in any other field they enter.” (Nurtured by Love) And he would know -- millions of children have become fine musicians through his Suzuki method, and gone on to successful and high paying careers in other areas.
My strong recommendation to parents is that they take the time to attend their scholar's performances and shows, carefully view their artwork, and show and give as much thoughtful praise as they can genuinely muster. Ask the scholar to talk about their experience and what the art work means to them. They will likely learn a lot about the profundity of meaning that the arts can go to, even as they draw closer to their teenaged daughter or son in discussing things that really matter to them. Yes, participating in the arts is making them happy, but why? Do they feel more connected with their peers in the group? Do they feel a sense of accomplishment in what they have created? Are they fascinated by the ideas and experiences that the art works brought up? Whenever possible, talk with them like an adult as they discuss the things they have learned. Maybe they can't even articulate any of these benefits, but you can start to make them out through their excited comments on the songs or drawings they have started doing non-stop at home. Yes they are happy, but not in the same way they are when they get to play video games all day. They are connected to their peers, to great artists from the past, and to great ideas and experiences that great minds have pondered over the centuries.
Also, we arts teachers can always use help facilitating concerts, shows, field trips, art exhibits, etc. Just ask.