Is it possible to pay a PCN (to London Borough of Barnet) with cash? No such option seems to be given on the PCN itself, but isn't cash considered 'legal tender' that cannot be refused in payment of a debt? Does that apply here?
IMO you can pay any PCN in cash. To force the issue, you would have to send it by Royal Mail Special Delivery to the address given on the PCN, and to put the matter beyond doubt I'd suggest you film yourself in a post office putting the cash in the envelope, sealing it and handing it over the counter and getting the receipt with the tracking number.
Once the cash is physically delivered into the council offices, it would be very hard for the council to argue that they've not been paid. If it's a parking regulation 9 PCN, I'd also suggest that they have no reason to seek the keeper's details from DVLA and if they do so anyway, that would be a breach of GDPR and would give the keeper grounds to sue.