You keep referring to "fines" and a "penalty". If you can find either of those words anywhere in all the correspondence, I will give you £100 for each occurrence.
You need to get it into your head that this is nothing to do with any fines, penalties or offences. It is a speculative invoice from an unregulated private parking company for an alleged breach of contract by the driver.
You actually have 33 days (not 28) from the date of the appeal rejection letter from PE to file your POPLA appeal. So, you have until 1st September to submit a POPLA appeal.
The 14 days you are referring to is PEs bribe discount period. Why are you even considering this? Maybe I should send you a speculative invoice for £100 and tell you that if you pay me £60 within 14 days I'll call it "quits".
If Saltburn Leisure Centre is the contractor of PE, then it may be worth reminding whoever is fobbing you off that they are jointly and severally liable for the actions of their agents and if they are happy that their patrons are invoiced for £100 by an unregulated private parking company who have accessed your personal data, they should be weary about their own liability. Ask them who is the monkey and who is the organ-grinder in their contractual relationship with Parking Eye.