Yes, I think that parents should be jailed when their children are arrested for drinking alcohol, because most of the time, a child who drinks alcohol has parents who looked the other way and didn't care that they were drinking. In some cases, it is the parent who provides the child the alcohol.
I have known a plethora of parents who allow their children to drink alcohol as long as they do it at home where they are safe. I do not believe parents should be jailed if their children are arrested for drinking alcohol. In this situation it sounds as though the teen is doing this outside of their parents permission. The offender should have to face the punishment, not the unaware parent.
I cannot offer blanket support for this proposal. We already have laws in place for people who supply alcohol to underage individuals. Simply enforce those laws, as is usually done. If the people who happen to supply the alcohol to minors are parents, then jail them. But just to arrest parents because their kids got drunk? Stupid idea.
As much as parents would like to, they can't watch their children at all times. Even moreso, they should allow their children to go out and explore the world, and if they start making mistakes at an early age, they can learn from it later on. Everyone needs to screw up once in a while.
A parent can only do so much sometimes their child is going to do it anyway even after telling them not to, I don't think the parents should get into legal trouble over something like this, if it happens repeatedly then some action could eventually be taken but it would have to happen very frequently.