In the article the kinematic problem for a new two-axis parallel manipulator with rotary hinged parallelogram, the source of which is a spindle unit, is performed. The analysis of the received kinematic dependencies is carried out. The presence of special provisions of the source body of the mechanism is established. The directions of further scientific researches are outlined. The parallel manipulator structure for consideration differs from previously developed fastenings of hinges of rods of variable length on rotary levers which rotation is coordinated with rotation of an initial link with a spindle unit. On the basis of the developed scheme mathematical dependences for lengths of actuators are established. The performed mathematical transformations allowed to solve direct and inverse kinematic problems for the mechanism. Analysis of the obtained kinematic dependences showed the presence of special positions of the initial body of the mechanism in the form of kinematic parallel singularities. Their graph is given. Graphs of changes of coordinates of an output link at change of lengths of rods of the drive of giving are shown. The analysis of graphic dependences shows existence of such combinations of constructive parameters of the mechanism at which the drive of giving starts to work in a multiplicative mode. The obtained dependences will allow to ensure the functioning of the kinematic circuits of the feed drive of this mechanism. The established points of the kinematic parallel singularity of the feed drive and its operation in the multiplicative mode require further consideration taking into account the dynamic loads. By selecting the design parameters of the mechanism, it is possible to achieve close to rectilinear relationships between changes in the length of the rods and the movement of the spindle unit.