Yorkshire picked up three of the five batting points available as they turned their overnight score of 292-5 into 433-9 declared.
Captain Andrew Gale’s innings was ended by Gary Keedy for 148, then Plunkett had some fun with an aggressive 51.
Surrey were 53-1 at stumps, nightwatchman Tim Linley protecting Kevin Pietersen and Vikram Solanki at the close.
Just eight overs were possible in a frustrating morning session that saw the early onset of rain and multiple delays as the umpires repeatedly assessed and changed the ball at Surrey’s insistence, much to the ire of the Yorkshire members.
Surrey nevertheless managed a wicket in that time, Linley finding Andrew Hodd’s edge to send him back for nine in the final over before an early lunch.
Gale was dropped by slip Solanki on 121 shortly after the restart, off Chris Tremlett.
It took Gale some time to reach the fluency he found on day one, but he was getting there when he fell two short of his 150.
It was the third over of Keedy’s spin and Gale misjudged a drive to be bowled.
Plunkett enjoyed himself with three fours in a single over off Tremlett, with five more helping him to a brisk half-century – a landmark he brought up with a short-arm pull off Jon Lewis.
The seamer got his own back a couple of overs later, Plunkett skying another attempted pull high and to slip, before Sidebottom (26) unveiled an unlikely array of sweeps to take the tally beyond 400.
Almost immediately, bad light and rain forced the players off again before another bright spell allowed Sidebottom and No 10 Steven Patterson to attack, swinging their bats at all comers.
Sidebottom found the boundary three times and hoisted Tremlett into the stands for six over mid-wicket. His innings ended when he attempted to thrash Linley down the ground only to be well caught by a back-pedalling Pietersen.
Yorkshire declared and another delay left only 18 overs to play, with Surrey openers Rory Burns (15no) and Arun Harinath (36) putting on 51 before the latter fell to Jack Brooks.