I want to focus on authoritarian and authoritative parenting, as these two styles really differ along that idea of punishment versus discipline. (The other two types of parents — permissive and neglectful — are both relatively low in control and socialization attempts.)
Permissive parenting is a type of parenting style characterized by low demands with high responsiveness. Permissive parents tend to be very loving, yet provide few guidelines and rules. These parents do not expect mature behavior from their children and often seem more like a friend than a parental figure.
Uninvolved parenting, sometimes referred to as neglectful parenting, is a style characterized by a lack of responsiveness to a child's needs. Uninvolved parents make few to no demands of their children and they are often indifferent, dismissive, or even completely neglectful.