You've been whalloped by the London-wide ban on off-carriageway parking in force since the 70s and which needs no signs or lines. Such bans can be enforced elsewhere in the UK but signs are needed to advise of a local ban.
For all PCNs issued by councils and Transport for London under several Acts, the owner of the vehicle has responsibility for payment or appealing PCNs. Having said this, the owner can authorise another person to appeal on his behalf, but if the appeal is lost, responsibility for payment stays with the owner.
All you can really do is submit representations based on your being from Birmingham, and were not familiar with the London ban. Ask for PCN cancellation for this first time offence as a goodwill gesture.