From memory of this its the registered keeper who is still liable under the byelaws. Please don't think this is the same as private car parks as TOCs can and do have ppcs act as their agents and they can use the bylaws to prosecute
Having had a casual refresh/browse of the 2005 Byelaws, I think Byelaw 14 is the only applicable one. This specifies that the
person in charge (
driver) of a vehicle must not fail to comply with signage, cause a hindrance, fail to comply with instructions, or fail to pay the appropriate fee for parking &c. - as per Byelaw 14(1), 14(2), 14(3) - breach of these rules can result in prosecution in the usual way and with the usual maximum penalties (Byelaw 24(1)).
Meanwhile, the
owner of a vehicle can be made to pay for costs incurred in clamping and/or removing and/or subsequently storing a vehicle left on property where the Byelaws apply (Byelaw 14(4)) (interestingly, the driver cannot but it would be somewhat of a moot point if the driver wants to recover/de-disable the vehicle). However, no provision is made for prosecuting the owner as far as I can tell, as there is not a rule that they 'shall' or 'shall not' do something, merely that they 'shall be liable', implying that there would be a civil debt.
Therefore, as far as I can tell, there are no legitimate grounds to obtain the Keeper's details from the DVLA, because being the Keeper of a vehicle is not the same as owning it, and the Byelaws specify that the driver or the owner is liable, not the Keeper. PoFA couldn't apply, both because the land is under statutory control (Byelaws) and because the owner or driver is liable, not the Keeper.
So unless the TOC/their agent know the identity of the driver, they have no way of prosecuting in the first place, and unless the vehicle has been clamped, removed and/or stored, there is no way the owner can be liable for anything as provision is only made for liability for these 'costs', not any parking charges. This is assuming they even know who the owner is, which, as stated, cannot be inferred merely from being the Keeper (for example, on a lease vehicle the lease company would be the owner but the lessee may be the Keeper).
So, because of the wording in the Byelaws and PoFA and their intersection, I can't see how the Keeper of a vehicle could be liable for any regular parking charges incurred on land under Byelaw control. On adjoining private land not falling under Byelaw control, as may often be the case for railway car parks, the Keeper could legitimately be pursued if PoFA, the relevant Accredited Trade Association's Code of Practice and all other laws (including planning permission, advertising consent, Consumer Rights Act, Consumer Contracts Regulations etc.) are adhered to, and if the contractual elements are acceptable (signage visible and sufficient, contractual terms not unenforceable e.g. by being forbidding or penalty being in small print).
Obviously, this is a high bar to pass, and while there are some PPCs that manage, or rather, bother, to pass it (eg ParkingEye), most others fail at one point or another, for lack of effort or purely thinking they don't have to fully comply. Hence parking charges in such car parks can be 'avoided' if controlling equipment like automatic barriers is not in operation. With Byelaw controlled land, you risk being clamped or towed and having to pay extortionate release fees, or the driver being prosecuted if identified, but neither the driver nor the owner could legitimately be pursued for any regular parking charges; the owner could be made to pay for clamping, towing or storage, and the driver could be prosecuted, but if the ToC/agent doesn't have the details then both will be impossible. Well, of course someone will have to pay release fees but it needn't be the owner.
Now, of course, I could be entirely wrong in this view, being untrained and unqualified in any legal capacity, but that is my reading of the situation (in England and Wales).