This isn’t as obvious an answer as you might think.
A mother automatically acquires parental responsibility on birth of a child.
A father who is married to the mother of a child at the time of the child’s birth also automatically acquires parental responsibility.
An unmarried father does not automatically acquire parental responsibility. If you are an unmarried father, you can acquire (or may have already acquired) parental responsibility in one of six ways: (a) by being registered as a child’s father on their birth certificate, (b) by entering into a parental responsibility agreement, (c) by applying to the court for a parental responsibility order, (d) by being appointed as a guardian over a child on the death of their mother, (e) if a child arrangements order is made in your favour or (f) if you marry your child’s mother.
Step-parents, (a person married/civil partners with one of the child’s natural parents) can obtain parental responsibility either by (a) obtaining a parental responsibility agreement (provided that any people who currently have parental responsibility over the child or children allow you to) and (b) if they don’t, by obtaining a parental responsibility order from the court.
If a parent is cohabiting with another person, that person isn’t considered to be a ‘Step-parent’ and so can’t apply for a parental responsibility order although they may be able to acquire it if a child arrangements order is made which states that the child should live with the cohabiting parent.