Thank you for your responses all. I have now read the forum guide.
I have the registered owner of the vehicle. I no longer have the initial letter from the company, but have attached the final reminder and debt collection letter that I have since received. It also appears that I have misunderstood what a consideration period is; I did not realise until now that the duration of the consideration period also needs to be paid for.

I appealed back in January (I do not have a copy of my exact appeal), on the basis that the machine was broken so it took me longer than usual to pay. I imagine that in this appeal I let them know that I was driving. I have included the response that I received on the 29th Jan at the end of this post.
I do not have photos of the car park signage, however here is a link to a google map view of the entrance. It is a small underground car park (space for perhaps 20 cars) accessible via a turning off a narrow street and down a ramp. Underground, there is 1 ticket machine at the bottom of the ramp.
https://www.google.co.uk/maps/place/Brick+St,+London/@51.5050322,-0.1489076,3a,60y,119.17h,90.93t/data=!3m7!1e1!3m5!1szvZuSBEXU8sxrICwZ9wuoQ!2e0!6shttps:%2F%2Fstreetviewpixels-pa.googleapis.com%2Fv1%2Fthumbnail%3Fcb_client%3Dmaps_sv.tactile%26w%3D900%26h%3D600%26pitch%3D-0.9292989480873928%26panoid%3DzvZuSBEXU8sxrICwZ9wuoQ%26yaw%3D119.16548072489734!7i16384!8i8192!4m6!3m5!1s0x4876052f4330180b:0x79b152b172b54ef2!8m2!3d51.5051696!4d-0.1483343!16s%2Fg%2F1tcxtqgh?entry=ttu&g_ep=EgoyMDI1MDQxNi4xIKXMDSoJLDEwMjExNjQwSAFQAw%3D%3DThanks in advanced for your continued help. Is it best for me just to suck it up and pay the increased charge?
Thank you for your appeal received on 11/01/2025 regarding the above detailed Parking Charge..
We have reviewed the case and considered the comments that you have made. This appeal has
been considered in conjunction with the photographs and any evidence provided. Our records
show that the notice was correctly issued as your vehicle was parked in breach of the Terms and
Conditions of the Car Park. The Parking Charge was issued to your vehicle because you failed to purchase a ticket to cover
your whole stay. It is the driver’s responsibility to ensure that they have read and understood the
terms and conditions for using the car park.British Parking Association Code of Practice - Consideration and Grace Periods
The driver must have the chance to consider the Terms and Conditions before entering into the
‘parking contract’ with you. If, having had that opportunity, the driver decides not to park but
chooses to leave the car park, you must provide them with a reasonable consideration period to
leave, before the driver can be bound by your parking contract. The amount of time in these
instances will vary dependant on site size and type. A consideration period shall not apply where a parking event takes place. (A ticket is purchased). Neither a consideration period or a grace period are periods of free parking and there is no
requirement for Britannia Parking to offer an additional allowance on top of a consideration or grace
period. We give motorists a 10 minute consideration period on arrival before entering into the ‘parking
contract’. If the driver has not purchased a ticket within this 10 minute period, a Parking Charge will
be issued.