This is misleading. I've had council tickets parking outside a hospital (with a disabled badge), because part of it is adopted by the council, but can appear to be the hospital. My crime was overstaying an hour, it was a traffic warden that got me, and the council that got paid (I appealed and to be fair I had no real defence as it wasn't life or death so had chance to move the car, and they allowed the 50% discount to remain for 14 days from their judgement).
Popla aren't independent because they're paid for by the PPCs so it's in their interest to keep the PPCs happy. The only upside is that their decisions are legally binding for the PPC, but not the alleged offender, so if they find against a motorist they're not obliged to pay, while a PPC has to deal with it they're found against. There's also more than one appeals body depending on which organisation the PPC signs up to. If they're a member of the British Parking Association, then yes, it is Popla. If the PPC is a member of the International Parking Community, then the appeal is handled by the Independent Appeals Service.
First things first:
AT NO POINT DO YOU STATE THE DRIVER WAS YOUR MOTHER! Always refer to her as the driver with no name given, and deny it was your mother. Unless they've got a clear picture of her in the car then they can't prove it. For all they know it could have been me driving. The only way they could force her to reveal who was driving is by what's known as a Norwich Pharmcal order, but there was a bit of a shake up a few years back, although I believe this is still the case. They won't got for one of these, because they're hard work and bloody expensive anyway.
Those pesky bays? They don't mean anything - they're effectively graffiti on the floor. On public roads you have to obey them, on private land they're legally unenforceable. Your mum could pull donuts in the car park and rag around at 90 mph and there's nothing they can do except tell her to leave. Speed limits, one way signs and any made up rule the land owner wants to make up are completely useless (although they sometimes do make life easier)
Do not contact the PPC, and do not give them more details (e-mail address, mobile number, etc). Your mum will get threatograms and other toilet paper from them for a bit, and keep the paper trail, but ignore. They'll eventually pass her across to the debt collection agency (next desk along) who will send more crap stating she owes money, and some of the shades of red they use are really pretty.
DCAs have no powers whatsoever. They may mention bailiffs in their letters, but they're not bailiffs. On the off chance you get a DCA that decides to doorstep, your mum should call the police if they don't leave immediately when told to. At this point your mum is perfectly allowed to let loose any large dogs she has, or even get a burly neighbour around to remove the DCA agent physically (remember: they have to be gentle, but occasionally accidents happen).
What you do need to keep an eye out for is stamped court papers. This is where it gets serious. If that happens I'd seriously recommend Consumer Action group (I'll link at the end). They'll explain what your mum needs to do, and how to do it. These guys know their stuff, even if they may disagree on finer points, so listen to their advice, do what they say, and they'll talk you through it. Most of this will be online and involve defence and evidence packs that need to be filed within a set period and whatever.
If it goes to court, remember to have it rerouted to your local courts. If you live far enough away, the PPCs usually won't turn up. I know it also depends on the judge, but they'll generally not take too kindly to hassling little old ladies (your mum needs to really work on the puppy dog eyes) for a debt that doesn't exist despite various rulings.
Whatever you do, don't pay up though, because they have no claim. There's no loss been suffered, and in the UK courts you're not allowed to go for penalties as a rule. The alleged contract is only an invitation to treat, and as there's no ability to negotiate it can't be forced (even via "if you don't like don't park"). Most of the little extras they use to bump the claim up are also considered business expenses and you can't sue someone for business expenses that would have been incurred in normal operation (they stopped doing some of them after a company dealing with "loss recovery" from shop lifters got slapped down for trying it on).
Seriously, it's just a legalised (and barely at that) scam.
A forum to discuss the legalities or unlawfulness of parking on private land
www.consumeractiongroup.co.uk