  The numerical simulation of the time-dependent Schr\"odinger equation for
quantum systems is a very active research topic. Yet, resolving the solution
sufficiently in space and time is challenging and mandates the use of modern
high-performance computing systems. While classical parallelization techniques
in space can reduce the runtime per time-step, novel parallel-in-time
integrators expose parallelism in the temporal domain. They work, however, not
very well for wave-type problems such as the Schr\"odinger equation. One
notable exception is the rational approximation of exponential integrators. In
this paper we derive an efficient variant of this approach suitable for the
complex-valued Schr\"odinger equation. Using the Faber-Carath\'eodory-Fej\'er
approximation, this variant is already a fast serial and in particular an
efficient time-parallel integrator. It can be used to augment classical
parallelization in space and we show the efficiency and effectiveness of our
method along the lines of two challenging, realistic examples.
