You say you have not "admitted who the driver was", but you do not "admit" anything. You simply decline to provide the drivers details as you are under no legal obligation to do so. Please show us exactly what you put in your appeal.
Did you send you proof of payment with your appeal? Show us their appeal rejection wording.
So, you have 33 days from the date of the appeal rejection to submit an appeal to POPLA. I suggest you have a search for other POPLA appeals on the forum to get an idea of how to put one together and what points to make.
When you have something, post it here and we will advise on any changes or edits you need to make. Don't send anything until we've had a chance to check it.
Initial Parking are not relying on PoFA to hold the Keeper liable and as long as they don't know the identity of the driver, they cannot hope to ever win this.