When your child does something wrong, your child should be made to understand why the action was wrong and potentially be punished for it. At a certain age, children should know the difference between right and wrong. If they do something wrong, then parents can be held liable for their actions, even though they didn't participate in a crime themselves.
What good does it do to hold a parent liable? In a criminal case, it might not be better than the alternative of sending a child to therapy or juvenile detention. Parental liability makes sense in a civil lawsuit, since the parents can be made to pay for damage a younger child causes, but in a criminal case, there's an argument to leave parents out of the case.
Parents can be held responsible for crimes due to laws against contributing to the delinquency of a minor. For example, if you have a gun in your home that your child obtains and uses for a crime, then you could be charged with a crime yourself. Parents may even be able to be charged for hacking or Internet crimes, since they are the ones who should be monitoring their children and are the ones who provide access to the Internet in their homes.
Normally, parents won't be held responsible for the actions of children under 8; children that age are unlikely to choose to participate in criminal actions. However, once they reach 8 to 10 years of age, parents can be held responsible until they reach the age of majority at up to 21 years old.
Source: FindLaw, "Parental Liability Basics," accessed Sep. 28, 2016