The first measurement charged hadron pseudorapidity and azimuthal angle distributions at low transverse momentum ($p_\mathrm{T}$) relative to the momentum vector of a $Z$ boson in lead-lead (\PbPb) collisions at a nucleon-nucleon center-of-mass energy $\sqrt{s_{NN}} = 5.02$ TeV is presented. By studying the hadrons produced opposite to the $Z$ boson, the analysis probes parton-medium interactions, including modifications to parton showers and medium response. Utilizing PbPb data from 2018 with an integrated luminosity of $1.67 \pm 0.03$ nb$^{-1}$ and proton-proton (pp) data from 2017 with $301 \pm 6$ pb$^{-1}$, the analysis examines the distributions in bins of charged-hadron $p_\mathrm{T}$. The analysis focuses on events containing at least one $Z$ boson with $40 < p_\mathrm{T} < 350$ GeV. A significant modification in the azimuthal angle and pseudorapidity distributions for charged hadrons in the range $1 < p_\mathrm{T} < 2$ GeV in PbPb collisions is observed when compared to reference measurements from pp collisions. The results are consistent with expectations from phenomenological models that include medium recoil, where the energy lost by the hard parton is transferred to the medium, and the medium-hole effect, which describes regions in the quark-gluon plasma depleted of energy due to parton-medium scattering. The data provide new information about the correlation between hard and soft particles in heavy ion collisions, which can be used to test predictions of various jet quenching models. Based on comparisons of PbPb data with pp references and predictions from theoretical models, this Letter presents the first evidence of medium-recoil and medium-hole effects caused by a hard probe.