Part-Time or Partial Homeschooling is a very broad term for parents taking charge of their kids' education. It doesn't mean that the family has left the public or private education system. You can even call it supplementing or tutoring.
For example, Mr. B goes to his Montessori Pre-School Prep Class three times per week and we home-school on Thursday and Friday. Another variation is when the child goes to school full time and the parent home-schools in the evening and/or on the weekend.
All children and families are different and it's impossible for the school system to meet everyone's unique learning needs. Many parents already partially home-school their kids without even thinking about it or calling it homeschooling.
Personally, we love Mr. B's school and feel that it's important for him to play with his friends, engage in arts and crafts and take a sports class on Monday. In addition, Mr. B has shown an eagerness to study some subjects on a deeper level than he would at school. So we would like to have our cake and eat it too. It's the best of both worlds. As long as Mr. B enjoys school he can go and have fun. We'll worry about meeting his personal learning needs at home by finding appropriate reading materials and activities to encourage his interests.