There has been case law on this subject, which I will set out - the £35 sounds excessive however if you did not pay, it would reach around £160 with debt collection charges - if they sued, they add on £35 for the court fee, £25 for a hearing fee and also interest so it can end up being well over £200 if you lost.
The term "fine" is a misnomer however, as it is not a criminal fine but a civil damages case. There is a case involving another parking company and a defendant named Mr Beavis.
The Beavis case concerned the initial £100 "fine", court fee and expenses.
SIP pursued you as they believe you are in breach of contract and are due damages.
The above case concerned the reasonableness of the fee (the amount) for an overstay. It hinged on whether the claimant parking company had actually lost that money due to the defendant's conduct.
In a free car park, there is no "loss" to speak of. In a paid car park, there is a loss as it is private property subject to parking restrictions.
It comes down to contract law. A parking fine results from breach of contract - either you didn’t pay when you used a car park, or you did but you overstayed. The fine is fairly low initially but will increase for as long as it remains unpaid - with debt collection charges so it can easily double.
As to whether the case applies to you, it may well do but each case is different and depends on its own facts. In a free car park (to a certain point) the damages would be little or nothing - I do not know if the car park you were in was free for a certain length of time or not. If is was, my view is you could challenge this fine.
I realise you have also paid the £35 now - my view is that this is not worth challenging.
I presume that the car park you were in was not free for a certain period of time as you mentioned using a tram - the car parks are usually paid from the moment you use it.