Thinking out loud... If the P&R was ever built, it would need to take into account the needs of those travelling from East Cleveland.
Would Nunthorpe be an attractive option for those in East Cleveland?

They have the option of the existing bus network (which has improved over the years) and the half hourly train service - which can take punters into the heart of Middlesbrough.
It can be easier driving to places like Saltburn and using the train in, than it is driving into Boro.
I am not sure a site near Nunthorpe would make the existing train or bus service any less attractive.