The best example of a home schooled child is Abraham Lincoln. Parents choose to homeschool their children for a variety of reasons. Some parents prefer for children to have lessons from a religious point of view. Evangelists will homeschool their children because of the travel their jobs entail.
Some parochial schools are too far away to attend, especially if one lives in a rural area.
Sometimes private schools are too expensive. Some children do not thrive well in a public school setting. Either the child is shy or easily distracted. The public school system may not be able to attend to the needs of these types of children. Parents may feel that the school system doesn't challenge their child or approve of the curriculum taught.