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Temporal cavity solitons (CSs) are pulses of light that can persist endlessly in dispersive, nonlinear optical resonators. They have been extensively studied in the context of resonators with purely cubic (Kerr-type) nonlinearity that are externally-driven with a monochromatic continuous wave laser -- in such systems, the solitons manifest themselves as unique attractors whose carrier frequency coincides with that of the external driving field. Recent experiments have, however, shown that a qualitatively different type of temporal CS can arise via parametric down-conversion in resonators with simultaneous quadratic and cubic nonlinearity. In contrast to conventional CSs in pure-Kerr resonators, these \emph{parametrically-driven cavity solitons} come in two different flavours with opposite phases, and they are spectrally centred at half of the frequency of the driving field. Here, we theoretically and numerically show that, under conditions of bichromatic driving, such parametrically-driven CSs can also arise in dispersive resonators with pure Kerr nonlinearity. In this case, the solitons arise through parametric four-wave mixing, come with two distinct phases, and have a carrier frequency in between the two external driving fields. We show that, when all waves are resonant, the solitons can experience long-range interactions due to their back-action on the intracavity fields at the pump frequencies, and we discuss the parameter requirements for the solitons' existence. Besides underlining the possibility of exciting a new type of cavity soliton in dispersive Kerr cavities, our work advances the theoretical modeling of resonators that are coherently-driven with polychromatic fields.