Parental responsibility is not the same as child custody.
Parental responsibility is where a parent has a legal duty and responsibility for a child.
This includes caring for them, and making decisions about their upbringing, health and educational needs. All mothers have parental responsibility but not all fathers do. A father that is married to the mother at the time of birth or is named on the birth certificate has legal parental responsibility.
On the other hand, child custody, now referred to as child arrangements are the living arrangements of the child including who they have contact with, where they go to school, and where they spend their time.
Those with parental responsibility are entitled to apply for child custody or child arrangements. But this does not mean that the court will allow the child to have contact with a parent who has parental responsibility. It is very much dependent on the situation and what is in the best interests of the child.