on the Virgin Trains East Coast page it doesn't show fares but Virgin operate a bus from York to Whitby leaving York at 08.26, 10.47 and 14.47.
It shows it as being operated by Virgin on the timetable too.
Well, as has been mentioned before on this thread, Priv discount is not valid for the bus link (in common with the majority of bus links) so don't be surprised when you don't get the discount.
That's right, but Priv is not a railcard and is not generally available on bus linksAll the rest of the railcards discount the fares if you put them in so find it a bit odd that a priv wouldn't. I'll ask next time I'm in town at the local station.
You can pick up the Coastliner at Malton station as well, and it's a spectacular run (especially if you can still get a bus that doesn't serve Goathland village and it's a double decker).
Double deckers run into Goathland down the steep hill so avoiding the low bridge, go to the church, turn round and go back the same way.