So Barnet Council are as useless as they always have been !
Well, the first thing to say is that the interval between your submission of informal reps and their response 5 months later is totally disgraceful and on its own likely to win an appeal at London Tribunals. In fact, with the limit on serving a Notice to Owner of 6 months from date of contravention, I think they will struggle to serve one to you because they cannot serve an NtO until 28 days have elapsed from the date of their letter.
So my advice is to NOT PAY, and wait for the Notice to Owner. This is sent to the name and address on the V5 Registration Certificate for the car. Is this you, and is it up-to-date and correct ?
Others here will no doubt comment.
Addendum
It would help if you could describe the circumstances leading up to the PCN being found on your car. For instance with my own circumstances, I used to take my mother, (in her late 90s) out and would park conveniently for her to walk a short distance to the location being visited using her Blue Badge. This sometimes required parking on yellow lines, single or double, so I assume you were doing something similar.