I suggest you ask them to provide evidence on the PCN he paid. Either there are two against his car, or he paid for somebody else's PCN by mistake.
And also ask for full details of the third PCN AF10804083, which they say was not paid and is still outstanding.
PCNs AF10804083 and AF101903675 (the PCN the payment was applied to) currently cannot be found on the Redbridge PCN portal.
AF10903686 is found but it says it's beyond the 28 day limit to submit representations. There doesn't seem to be any accessible photos. How did you get them?
However the elephant in the room is the Blue Badge. By virtue of display of the BB father is entitled to a three hour exemption from any yellow line restrictions, providing that there were no loading restrictions in-force at the time of parking. The photos don't seem to show any yellow kerb dashes so that means no loading restrictions applied. You say father parked around 1pm and the first photo is timed at 13:06:42. So the three hours has barely started.
Is the sign in the photos on the same side of the road as where father parked?
Was the car parked to the right or the left of the sign?
Why didn't you, or rather your father, mention the BB in the formal representation against the Notice to Owner? Or did you?
Was the time on the BB clock set correctly? From the photos I can just about make out that there is a clock but not the time that's set. Do you have any clearer photos of the BB and clock as displayed?
And yes the letter does seem to imply that £80 has been applied against the PCN referenced in the NTO and at the top of the Notice of Rejection. The amounts due if you lose at Adjudication or if a Charge Certificate is issued are incorrect. They can only be reconciled if a part payment of £80 has been applied.