Certainly parents can install parental control software, but what does this have to do with children's PII being shared with sites?
The parent can block or just never approve all the sites that require PII.
but it's not PII as usually regarded
We will never agree here. All the companies I worked for financial considered any attribute of the person to be PII, even their IP address. We were audited very strictly on this. If a users age was disclosed to a third party without their written consent that was a contract violation and came with severe monetary penalties. Parents should expect this to be the minimum standard. It's their children, not the corporation or governments children.