That's not how footway parking works. It is banned everywhere except where signs say otherwise. This is a cul-de-sac, so you have to drive past that sign on the way into the road. Once you're in an exempt area, there is no requirement to have a sign for each bay, as the footway parking exemption (with any qualification such as "in marked bays") applies to the whole street.
This is the best I can come up with in the circumstances:
Dear London Borough of Barnet,
In the first instance I would invite you to cancel the penalty charge on a discretionary basis, given that I parked at night and the contravention appears to be minimal if not de-minimis.
If you are not minded to do so, I would ask that you please provide a copy of the footway parking resolution duly made under section 15(4) of the Greater London Council (General Powers) Act 1974, so that I may verify whether the council has discharged its duties under section 15(5).
Yours faithfully,