Following shape optimisation approaches taking into account PDE-constraints as met in industrial engineering yields to adjoint PDE-systems being unilaterally coupled to the primal PDE-system to be solved. Focusing on unsteady problems, a major drawback consists today in massive requirements with regard to computational ressources and overall computational run times. However, as the tendency towards reaching the limits of transistor performance is confirmed and increasingly limiting the potential of numerical applications in science and industry, the efficient usage of highly distributed and parallelised numerical algorithms has become essential in order to speed up run times of large simulations. When facing coupled PDE-systems, formulating efficient, fast and robust numerical algorithms becomes even more important and challenging though. % In the submitted research, a parallel-in-time formulation of the adjoint incompressible unsteady Reynolds-averaged Navier-Stokes (URANS) equations is presented. Achieving a parallelisation in time for transient shape optimisation problems has obtained little attention so far when compared to algorithmic parallelisation of linear algebra problems or a parallelisation along spatial dimensions. However, as these algorithms tend to reach a saturated degree of parallelisation as well, additional levels of parallelisation may become attractive even while suffering from a comparably low parallelisation efficiency. % In this research, the parallelisation along the time dimension is imposed on the unilaterally coupled problem of both primal and adjoint URANS equations with the goal to achieve faster approximations of the resulting gradient in each steepest descent iteration when applied to shape optimization problems of compressible channel flows. This is achieved by relying on a multiple-shooting framework as proposed in [1,2] that allows to maintain the same time-stepping and the same spatial mesh structures as used in a serial-in-time consideration of the transient problem. The focus of the presented research is on a discussion how the reversion of the time-axis and the coupling between primal and adjoint URANS problem affect the application of time-parallel integration schemes and to which extent the computation of the shape gradient in each optimisation iterate can be accelerated.