Are adult children financially liable for elderly parents auto accident?

Does the parent have auto insurance? Was the parent driving the adult child's car with or without permission?

The only theory that I know of upon which the children would be responsible would be what is called "negligent entrustment". If the child reasonably should have known that the parent, when behind the wheel, would be an unreasonable danger to self or others, and nevertheless let the parent use the child's car, maybe.

In general if an adult family member has an accident the other members of the family are not liable. There are exceptions, as AK_lawyer pointed out. And as WG is alluding, if a person gets into an accident with a borrowed car things can get dicey. In general the car owner's collision insurance will cover collision damage, though the owner will have to pay the deductible. As for liability insurance - generally the driver's policy is primary, but if his limits are exceeded the owner's insurance may have to step in as secondary insurer. If the driver is uninsured then the owner is responsible for damages. This assumes the owner gave permission for the driver to borrow the car.

... If the driver is uninsured then the owner is responsible for damages. ...

Only if that's what the mandatory insurance law in the jurisdiction provides.

Otherwise, as I indicated, only if the owner did something negligent (such as lending the car to someone who is known to be a terrible driver) will the owner be liable.

Too many "maybe" here in China, yes they would be.

Whose car?
Are the elderly parents too confused to be driving alone?

I am not liable for you, driving, but if we were together, and I knew you were drunk, and I allowed you to drive, in some places, I may be liable.

So you need to explain this, much better, to even have a guess at your answer.