No, it’s really not the same thing. You can legislate better schools with a variety of methods, the main point being that you’re regulating government jobs(to oversimplify). You’re more limited to negative legislation for parents, such as punishing child abuse. I guess you could technically legislate certain mandates for parents to be better parents, but like, good luck passing said legislation. And even if you do(and this is the big boi), how the fuck do you enforce that??? And on top of even that, how can you be sure parents will be qualified/able to teach their kids such a wide variety of skills? You can fire teachers for incompetence and publicly investigate school districts for failing to faithfully implement good practice. And it should also be mentioned that shifting these expectations (especially via legislation) onto parents will disproportionately burden the poor who will be less likely to have the time, skills, or knowledge to teach said things.
Oh yeah, I guess those can also do that.