This talk presents the development of parallel-in-time solution strategies for the three-dimensional incompressible Navier-Stokes equations, which exploit discretely divergence-free finite elements to eliminate the saddle-point structure typically encountered in mixed formulations. Starting with the well-known Q2-P1 finite element pair, we derive vector-valued shape functions that automatically satisfy the discrete incompressibility condition. When these specific functions are used to discretize the Navier-Stokes equations, all contributions associated with the pressure term cancel out, resulting in systems that involve only velocity variables. The absence of the zero block characteristic for mixed discretizations allows for the use of a wide range of preconditioners and leads to highly efficient iterative solvers based on geometric multigrid techniques. After introducing the key properties and challenges of discretely divergence-free finite elements in three dimensions, we discuss their integration into parallel-in-time algorithms. A major advantage of this formulation is that no Schur complement techniques are required, enabling very efficient line smoothers within multigrid waveform relaxation techniques. Furthermore, this framework facilitates the use of modern hardware architectures with extensive vector unit parallelism by employing time averaging strategies within space-time multigrid schemes. Finally, various numerical experiments are presented that demonstrate the effectiveness of this design methodology for solving the incompressible Navier-Stokes equations and outline promising directions for future research.