For liability for anyone's negligence, you have to have control over that person.
Therefore, employer would be responsible for damages caused by an employee during the conduct of employer's business, such as in the business of delivery, installation, repair etc.
To hold a parent liable, the plaintiff has to show that the parent has control over the adult child. Now, when the adult child was acting on behalf of the parent, such as picking up siblings, the adult child could be treated as the parent's employee (or agent 代理）. so in this limited situation, the parent might be liable for harm caused during the conduct of "family business".
If the child just harms someone on his/her own, usually the parents won't be liable.
但是谁也不敢打包票，谁知道你和孩子会不会no zuo no die?
