If heard on the papers then the adjudicator could only go on written evidence, snd IMO yours is deficient unless there's something which proves to the required standard that the author of the permission has the legal status required to grant the exemption.
I have not read any tribunal decisions on this point so don't know what an adjudicator would need. The words of the Act are:
3)The second exception is where the vehicle is parked outside residential premises by or with the consent (but not consent given for reward) of the occupier of the premises.
This exception does not apply in the case of a shared driveway.
No mention of consent in writing, even less written consent in advance. But you knowing and acting upon such permission is one thing, convincing an arbiter is another. This is an appeal and absent the occupier presenting oral evidence you're left with what's written. What supplementary questions might the adjudicator have in their head which you might not have anticipated in your written submissions?
On the whole we favour personal hearings.