Kids are going to be kids. Sometimes that means they're going to make a mess in the kitchen, and at other times, that means parents are going to get a call in the middle of the night from the police. Most of the time, when a kid commits a criminal act, the consequences will be theirs to face as very few states impose any sort of parental criminal liability.
However, when it is discovered that a child has been using drugs, a parent could face criminal liability depending on the situation. Generally, so long as the parent was not supplying those drugs, knowingly or otherwise, or did not know about the child's drug use, nor encourage it, a parent does not have to worry about being arrested or facing criminal liability. On the other hand, when children are exposed to drugs because of their parents, parents can be arrested and face criminal charges.
Good Parent Laws
While there is no law that requires a parent to be a "good" parent, there are laws that criminalize bad parenting that results in child abuse or neglect, as well as the unfortunately all too common crime of contributing to the delinquency of a minor. If a child's drug use is found to be the result of child neglect, or worse, a parent's encouragement, arrest and criminal charges against the parent are likely to follow.
Also, if a parent is a drug user (alcohol, cigarettes, and marijuana are all drugs by the way), and their child is able to access their parent's drugs, that can also form the basis of criminal liability for child endangerment, as well as potentially other crimes. This occurs most commonly when children get into their parents liquor cabinet, cigarettes, or even now, legal marijuana, and start serving friends marijuana, cigarettes and alcohol. In situations such as these, a parent can be arrested even if their child has an drug fueled party they are unaware of.
Child Causes Damages or Injury to Others
A parent can also be held liable civilly, as well as criminally, for the monetary damages that their child is found liable for. This is common when an underage driver causes an accident due to DUI.