Continuum parallel robots are a new class of parallel robots in which the end-effector is moved by controlling the deformations of their flexible legs. In general, all degrees of freedom of the robot platform are coupled, leading to the necessity to have the same number of motors as the number of degrees of freedom offered by the space of displacements. Recent works showed that it is possible to design continuum parallel robots such that the orientation of their platform is constrained to be constant. Planar designs able to obtain this interesting performance are made with the use of flexible parallelograms. However, a single design has been studied which presents legs’ overlap. There is a lack of a deeper study of the other possible designs able to obtain the same types of mobilities while being more practical to build. In the present paper, we thus explore how different platform shapes influence the angular deviations of the platform as well as the robot’s workspace.