In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overheard come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we introduce a new strategy, Communication Aware Adaptive Parareal (CAAP) to overcome some of the challenges. With CAAP, we choose time-subdomains short enough that convergence of the Parareal algorithm is quick, yet long enough that the overheard of communicating time-subdomain interfaces does not induce a new limit to parallel speed-up. Furthermore, we propose an adaptive work scheduling algorithm that overlaps consecutive Parareal cycles and decouples the number of time-subdomains and number of active node-groups in an efficient manner to allow for comparatively high parallel eciency. We demonstrate the viability of CAAP trough the parallel-in-time integration of a hyperbolic system of PDEs in the form of the two-dimensional nonlinear shallow-water wave equation solved using a 3rd order accurate WENO-RK discretization. For the computational cheap approximate operator needed as a preconditioner in the Parareal corrections we use a lower order Roe type discretization. Time-parallel integration of purely hyperbolic type evolution problems is traditionally considered impractical. Trough large-scale numerical experiments we demonstrate that with CAAP, it is possible not only to obtain time-parallel speedup on this class of evolution problems, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. The approach is widely applicable for parallel-in-time integration over long time domains, regardless of the class of evolution problem.