The electric shuttle bus used to run before the days of low cost it was replaced by a service bus for a short while. Then the green fence with the locked gates appeared to make it difficult to park anywhere other than the official carpark. A subsidised bus did run between Darlington Station and the airport for a period, no idea who paid but bet it wasn't Peel.
There is a service bus again to Darlington, it doesn't reach the terminal and again the times are poor, last bus something like 4.30pm.