High-energy photonuclear ($\gamma +A$) scattering in ultra-peripheral heavy-ion collisions provides a unique probe of nuclear structure. This Letter studies the dependence of $\gamma +A$ jet production in ultra-peripheral Pb+Pb collisions at $\sqrt{s_{{}_{\mathrm{\text{NN}}}}}=5.02$ TeV on the presence of forward neutron emission from either nucleus. The data was taken in 2018 with the ATLAS detector at the LHC and corresponds to an integrated luminosity of $1.72$ nb${}^{-1}$. The kinematics of the hard $\gamma +A$ processes, expressed via the particle-level photon ($z_{-}$) or nuclear parton ($x_{+}$) momentum fractions, are determined from $R=0.4$ jets reconstructed using the anti-$k_t$ algorithm. At lower $z_{-}$, where the non-diffractive component dominates, the nuclear parton distribution can be cleanly probed in collisions that leave the struck nucleus essentially intact. Such collisions are expected to probe larger impact parameters ($b_{\mathrm{\text{A}}}$) within the target. The shape of the $\gamma +A$ cross-section as a function of $x_{+}$ in such collisions is found to differ from that in $\gamma +A$ collisions accompanied by forward neutron emission, with an observed significance of $6.0\sigma$. These results are consistent at large $x_{+}$ with large $b_{\mathrm{\text{A}}}$ collisions exhibiting no modifications to the parton distributions that are usually observed in hard scattering processes involving nuclei, relative to collisions with smaller $b_{\mathrm{\text{A}}}$. Thus, these measurements provide an experimental observation that the modifications to nuclear parton distributions vary with impact parameter.