Large-scale optimization problems governed by partial differential equations (PDEs) occur in a multitude of applications, for example in inverse problems for non-destructive testing of materials and structures, in flow-control problems, or in individualized medicine. Algorithms for the numerical solution of such PDE-constrained optimization problems are computationally extremely demanding, as they typically require multiple PDE solves during the iterative optimization process. This is especially challenging for transient problems. In order to tackle real-life applications, it is not only essential to devise efficient discretization schemes, but also to use advanced techniques to exploit computer architectures and decrease the time-to-solution, which otherwise is prohibitively long One approach is to utilize the increasing number of compute cores available in current HPC clusters. In addition to more common spatial parallelization, time-parallel methods are receiving increasing interest in the last decade. There, iterative multilevel schemes such as Parareal, MGRIT and PFASST are currently state of the art and achieve notable parallel efficiency, but work especially well for parabolic PDEs. Other approaches like ParaDiag work without coarsening in space or time, and can achieve significant speedups also for hyperbolic problems. In this talk we will discuss how time-parallel time-integration methods can be leveraged for computationally challenging inverse problems. We describe algorithmic approaches and challenges, and investigate performance for two application problems: (i) bathymetry reconstruction for the shallow water equations, and (ii) estimation of the motion of contrast agents from 3d dynamic ultrasound measurements.