The OP's parking created exactly the nuisance the law was introduced to penalise and this is indisputable IMO. The nuisance is clear for everyone to see.
I have said that they should succeed, but they need to acknowledge the nuisance they created.
My other references were simple. The power of an authority to disapply the prohibition is as follows:
4)A highway authority may by resolution, or in the case of the Secretary of State by such notice as appears to him to be appropriate, authorise, from a date specified in the resolution or notice, the parking of vehicles[F7—
(a)]on, or [F8on or over a road or part of a road] which is a highway other than a carriageway[F
No mention of wheels, which occurs in the prohibition, which is odd because surely vehicles are already permitted to park 'on or over' provided that their wheels aren't 'on or over'. As you can see, these are not exactly complementary and I simply posed the question - in fact a warning- to the OP to tread carefully. Make their point, but not in a finger-wagging manner.
Adjudicators can be fickle.