Answer:
The only qualification a parent needs is that they be the custodial parent of the child who is to be homeschooled. However, if the child receives instruction from someone other than the parent, that person needs to be a certified child educator. The same thing applies if a parent who is homeschooling their children decides to teach other children as well...the parent who is teaching those additional children would then need to have a teaching certificate.
Answer
Qualifications vary depending on the legislation in each state. The best resource that I have found is The Homeschool Legal Defense Association: http://www.hslda.org/.