Have you seen the timetable for Tees-side Airport Railway station?
Sundays only if I recall correctly:
Realtime Trains | Departures from Tees-side Airport
Not really a viable alternative to car parking no matter what the cost! Looking at arrivals and departures, it looks like it could work if you wanted to fly Loganair to Norwich on a Sunday, and return from Amsterdam by KLM the next Sunday...