British Birth Certificates have been issued in many different forms over the years. Within England & Wales, there were two common formats, usually known as either a Long-Form or Short-Form certificate. Because the Long-Form version (the one that includes parent(s) information) is physically the same size as an A4 piece of paper, it has come to be known as an A4 version.
For the avoidance of doubt, if you have been asked to present an A4 version of your birth certificate, this simply means a version that includes the parents of the child stated on the certificate. This is now common practice to assist in preventing identity fraud and to prove entitlement to British Citizenship.
If you only have a short copy of your birth certificate, you can purchase an A4 version here.