As a starting point, in general, with private parking charges, it is the driver who is liable - the driver is the person who, through their conduct, entered any contract offered by the signage at the site. There are mechanisms through which parking companies can recover unpaid parking charges from the hirer or registered keeper of a vehicle, if they do not know who was driving, but the starting point is always the driver. In a case of a driver under instruction, the person behind the wheel is still the driver. (That said, as an aside, I would expect a competent driving instructor to be aware of the requirement to note any terms and conditions when entering private land, and avoid taking a student somewhere where they may become liable for a charge - and if they subsequently did, I'd suggest the decent thing to do would be for the instructor to pay. This is my opinion rather than a legal position, though.)
However, in this case, your instructor has chosen to pay the parking charge, thereby effectively accepting that he owes the money. If he wanted to hold you liable for the charge, he could have provided the parking company with your name and address, as the notice he received will have made clear to him. This would have removed his liability entirely, but instead he made the choice to pay. The fact he's threatening you with 'debt collectors' suggests he may be chancing his arm. He'd need to sue you in the County Court if he wanted to recover the money, and you would seem to have the makings of a decent defence... He voluntarily paid an invoice issued to him by another company, if he believed you were liable, he could have told them that.
Has he sent you copies of the parking charge notices?