Hyperspectral imaging has a wide range of applications relying on remote material identification, including astronomy, mineralogy, and agriculture; however, due to the large volume of data involved, the complexity and cost of hyperspectral imagers can be prohibitive. The exploitation of redundancies along the spatial and spectral dimensions of a hyperspectral image of a scene has created new paradigms that overcome the limitations of traditional imaging systems. While compressive sensing (CS) approaches have been proposed and simulated with success on already acquired hyperspectral imagery, most of the existing work relies on the capability to simultaneously measure the spatial and spectral dimensions of the hyperspectral cube. Most real-life devices, however, are limited to sampling one or two dimensions at a time, which renders a significant portion of the existing work unfeasible. We propose a new variant of the recently proposed serial hybrid vectorial and tensorial compressive sensing (HCS-S) algorithm that, like its predecessor, is compatible with real-life devices both in terms of the acquisition and reconstruction requirements. The newly introduced approach is parallelizable, and we abbreviate it as HCS-P. Together, HCS-S and HCS-P comprise a generalized framework for hybrid tenso-vectorial compressive sensing, or HCS for short. We perform a detailed analysis that demonstrates the uniqueness of the signal reconstructed by both the original HCS-S and the proposed HCS-P algorithms. Last, we analyze the behavior of the HCS reconstruction algorithms in the presence of measurement noise, both theoretically and experimentally.