The tunneling path between the CuO2 layers in cuprate superconductors and a scanning-tunneling-microscope tip passes through a barrier made from other oxide layers. This opens up the possibility that inelastic processes in the barrier contribute to the tunneling spectra. Such processes cause one or possibly more peaks in the second derivative current-voltage spectra displaced by phonon energies from the density of states singularity associated with superconductivity. Calculations of inelastic processes generated by apical O phonons show good qualitative agreement with recent experiments reported by Lee et al. Further tests to discriminate between these inelastic processes and coupling to planar phonons are proposed.