Certainly parents can install parental control software, but what does this have to do with children's PII being shared with sites?
Just so we're on the same page, the way AB1043 works is that the OS determines the user's age and then shares the age bracket with apps. No PII is shared with sites (this is not to say that the age isn't sensitive, but it's not PII as usually regarded). Is your concern here that sites get access to children's information because children visit certain sites regardless of legislation? That's a real thing, but it seems mostly orthogonal to age assurance.