Disclosed is a two-layer three-rail planar robot with a parallelogram, including a fixed platform, a moving platform, and three branched chains. Three planar curved rails I are provided on the fixed platform. Three planar curved rails II are fixedly connected to the moving platform. Each planar curved rail I is connected to a planar curved rail II corresponding to the planar curved rail I by one of the branched chains. Each of the branched chains includes a slider I, two connecting rods provided in parallel, a slider II. The slider I is slidably connected to the planar curved rail I. The slider I is rotatably connected to one end of each connecting rod by a revolute pair I, the other end of the connecting rod is rotatably connected to the slider II by a revolute pair II. The slider II is slidably connected to the planar curved rail II.