In a society with significant child labor, it's not clear to me how "just pay the parents more" is a serious response. When significant child labor exists, it does so for a reason.
Given that parents would probably prefer their kids get an education instead of working, it's not clear to me that depriving them of the option to work in a factory will result in the child going to school. In all likelihood, the child will continue to work, albeit at their next-best (perhaps illicit) option; their best option (via revealed preference) having been removed by enlightened westerners who don't have to face the same constrained set of choices.
Let's look, amorally, at why China has a serious interest in not having child labor.
1. It cheapens their labor, and trades an adult's job for a child. Most of the money for the jobs with potential child labor is coming from out of the country; why not soak them for a few more Yuan per person?
2. It upsets their primary business partners and reinforces the "China regularly violates human rights" stereotype that the West uses as a cudgel.
3. It trades a worker's future gains for much less siginificant short-term gains. China wants more highly-skilled jobs for its young people. It wants more Chinese luxury goods, and wants them sold the world 'round. Every child working in a factor is another child that isn't getting the education to make China a bigger world power and take the reins of a given industry.
All of which is irrelevant to the fact that the best option for many children (again, revealed preference) is to work, and depriving them of that option is likely to lead to a net worse condition.
But to your point, I'm quite certain that "China" would prefer to have an economy sufficiently productive and wealthy that child labor is no longer a choice parents feel they need to make. Banning the effect does not change the cause.
"When significant child labor exists, it does so for a reason."
The reason is that it is in the interest of wealthy factory owners to keep the cost of labor as low as possible, just like when child labor was prevalent in the United States.
You might want to look into why it was possible for child labor to end, as well. Every economy that has transitioned from rural/agricultural to urban/industrial has gone through a period of "exploitative" labor practices. I'm not sure it's necessarily inevitable, but it seems to be the norm.
Lower costs explain why child labor is in demand, but says nothing about supply, namely why parents send their children off to work in a factory, something parents in productive, wealthy economies are unlikely to do.
Which is why it's important to give parents a working wage.
Part of that is strict laws about child labour, to avoid factory owners exploiting poor people. Creating a level field makes it easier for ethical employers to follow their ethics.
It would be nice if some of the premium paid for some products in the west was used to pay poor workers better, or used for health and educational projects.
Given that parents would probably prefer their kids get an education instead of working, it's not clear to me that depriving them of the option to work in a factory will result in the child going to school. In all likelihood, the child will continue to work, albeit at their next-best (perhaps illicit) option; their best option (via revealed preference) having been removed by enlightened westerners who don't have to face the same constrained set of choices.