Tunable x-ray radiation sources are of wide importance for imaging and spectroscopy in fundamental science, medicine, and industry. The growing demand for highly tunable, high-brightness lab-scale x-ray sources motivates research of new mechanisms of x-ray generation. Parametric x-ray radiation (PXR) is a mechanism for tunable x-ray radiation from free electrons traversing crystalline materials. Although PXR has been investigated over decades, it remained limited in usages due to the low flux and strict dependence on fixed crystal properties. Here, we find new effects hiding in the PXR mechanisms, which provide control over the radiation polarization and spatial and temporal distribution. The radiation can form ultrashort pulses and delta-pulse trains, which makes the new effects fundamentally different from all conventional mechanisms of x-ray generation. We show how these new effects can be created from free-electron interactions with van der Waals materials. Furthermore, we consider free electrons traversing near material edges, which provides an additional degree of tunability in angular distribution and polarization of PXR. Our findings enable us to utilize recent breakthroughs in the atomic-scale design of 2D material heterostructures to provide platforms for creating tunable x-ray pulses.