They lie - in some states a HS parent has to have a high school diploma or equivelant; but in most it's not even a requirement. Otherwise, in some states, the child has to be evaluated by a certified teacher (one with a degree in education) but the HS parent is not required to have a college degree.
This is one of those little digs that is usually used as a deterant to keep parents from HS'ing. This is another argument that cracks me up because they are essentially saying - "Wait, you've gone through PS? No, no NO! You are not equiped to teach a child anything!!!" Which is funny because the reason for many of us to choose HS is because of the sub-standard levels of PS education. heheheh.
So I suggest googling - homeschool laws in _____ and you'll see what is required of you.
You need to check your state laws for surprise requirements, but as far as I know NO state has a minimum education level required for the parent. Studies have shown that the parent's education level doesn't affect the child's success, and they can't measure your self taught learning.
No, you don't. Private schools with hundreds of kids very often don't even have to have college educated teachers on staff. They can hire the person they desire to teach children. Homeschools have different requirements in every state, but I have never heard of any state requiring a college degree for any age level teaching.