Quantum mechanics places significant restrictions on the hydrodynamics of superfluid flows. Despite this it has been observed that turbulence in superfluids can, in a statistical sense, share many of the properties of its classical brethren; coherent bundles of superfluid vortices are often invoked as an important feature leading to this quasiclassical behavior. A recent experimental study [E. Rusaouen, B. Rousset, and P.-E. Roche, Europhys. Lett. 118, 14005, (2017)10.1209/0295-5075/118/14005] inferred the presence of these bundles through intermittency in the pressure field; however, direct visualization of the quantized vortices to corroborate this finding was not possible. In this work, we performed detailed numerical simulations of superfluid turbulence at the level of individual quantized vortices through the vortex filament model. Through course graining of the turbulent fields, we find compelling evidence supporting these conclusions at low temperature. Moreover, elementary simulations of an isolated bundle show that the number of vortices inside a bundle can be directly inferred from the magnitude of the pressure dip, with good theoretical agreement derived from the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations. Full simulations of superfluid turbulence show strong spatial correlations between course-grained vorticity and low-pressure regions, with intermittent vortex bundles appearing as deviations from the underlying Maxwellian (vorticity) and Gaussian (pressure) distributions. Finally, simulations of a decaying random tangle in an ultraquantum regime show a unique fingerprint in the evolution of the pressure distribution, which we argue can be fully understood using the HVBK framework.