Was in York on Saturday, so whilst youngest son was viewing the university, I toddled round to The Posterior Gate to view the facilities. It has fairly extensive outside, partly covered, areas where bikes should not be much of a problem. I suspect these areas are for summertime drinking but I don't think food is allowed. It backs on to the river and a large number of bikes could be parked on the riverside decking/terrace which would otherwise be empty.
Coming from the South/Fulford the middle of the road is barriered at the point where you need to turn right to get to the main entrance. There is a gap for pedestrians, so I would suggest pulling in and crossing the road on foot, otherwise you can go on and use the roundabout to retrace on the correct side of the barrier.
Did not stay to eat or drink - I can only face a 'spoons breakfast after about 400k... but it was not busy.