All parents are "home school parents". Some choose to share that educational mission with others outside of the home as their children mature. When they do, those partners in the educational mission should look first to the parents, not to a bloated federal bureaucracy.
If you "share the educational mission" by sending your child off to pubic or private school you are not a homeschooler.
A homeschool parent chooses the curriculum for their children, chooses the textbooks, creates lesson plans, grades papers, gives test, organizes field trips, and MEETS all the legal requirements the state they live in has for homeschoolers.
If you merely help your child with homework and "supplement" what they learn in public or private school you are NOT a homeschooler and it is insulting to homeschoolers to imply you are. It also gives ANTI-homeschoolers ammunition to use against homeschoolers. After all if Keith Public School Parent is a homeschooler and sends his child off to PUBLIC SCHOOL why can't all homeschool parents send their kids to public school and just homeschool after school like Keith Public School Parent? We then have to waste time explaining to these imbeciles that know nothing about homeschooling that Keith Public School Parent isn't a homeschooler even though he is stupidly insisting on calling himself one, because his God says something about parents teaching their kids in the Bible.
And yes, to some extent all parents teach their children. We teach them how to do many things before they are school age and we hopefully continue to influence them throughout their lives regardless of the educational choices we make for them. And yes, no matter what educational choice we make for our children we as parents should be involved in their education. But if you send your children to public or private school you are not a homeschooler.