Parametrically driven cavity solitons (PDCS), unlike single-pumped cavity solitons, are localized optical pulses arising from bichromatic parametric drive. These cavity solitons, recently discovered in pure-Kerr media, offer great promise for nonlinear dynamics studies and metrology. Here, we present a multimode quantum description of pure-Kerr PDCS. In the below threshold regime, we verify single- and two-mode squeezing, while above threshold we uncover novel ``quantum" dispersive waves—the quantum analog of soliton Cherenkov radiation. Besides revealing these unexplored quantum properties, we show that PDCS generates up to 20 dB of squeezing, only limited by coupling and intrinsic losses for experimentally routine parameters. We therefore provide a pathway to observe strong multimode quantum noise reduction in these systems.