We study the interaction of a laser cavity-soliton microcomb with an externally coupled, co-propagating tunable CW pump, observing parametric Kerr interactions which lead to the formation of both a cross-phase modulation and a four-wave mixing replica of the laser cavity-soliton. We compare and explain the dependence of the microcomb spectra from both the cavity-soliton and pump parameters, demonstrating the ability to adjust the microcomb externally without breaking or interfering with the soliton state. The parametric nature of the process agrees with numerical simulations. The parametric extended state maintains the typical robustness of laser-cavity solitons.