Yorkshire took one Durham wicket during a rain-interrupted afternoon session on day two at Headingley.
Only 11.4 overs were possible since lunch thanks to two weather delays, the last of which came shortly after 3pm.
There was no play after this and Durham were 50-1 in the 21st over at tea after the hosts had been dismissed for 426.
Tim Bresnan took the only wicket to fall when he had Mark Stoneman caught behind by Jonny Bairstow for 21, leaving the score at 39-1.
Scott Borthwick was handed a life when Adam Lyth dropped a chance at second slip off Bresnan shortly before the first rain delay at 2.20pm with Durham on 49-1.