Yes: for example –
Parents who are married both have parental responsibility.
A local authority is given parental responsibility when a care order is made but the mother (and maybe the father) will continue to have parental responsibility along with the local authority. The local authority will be in the driving seat, however, and can override the wishes of the parents if they cannot reach an agreement. The same applies once a placement order is made: the local authority can determine the extent to which a parent or guardian or prospective adopters may exercise parental responsibility. At this point, the local authority, parents / guardians and the prospective adopters all share parental responsibility.
A grandmother may have an order saying a child should live with her so she gets parental responsibility. The mother (and maybe the father) will still have parental responsibility.
A Special Guardian will share parental responsibility with the parents but can generally exercise it to the exclusion of anyone else. An exception applies in circumstances where the law provides that the consent of more than one person with PR is required (for example, the sterilisation of a child). The Special Guardian may not give consent to adoption or placement for adoption and may not change the child’s surname or take them out of the UK for longer than 3 months without the consent of everyone with PR or permission from the court.
Where more than one person has PR each of them may act alone in meeting that responsibility unless the law specifically requires joint agreement such as over a change of surname.