Strong radial confinement in semiconductor nanowires leads to modified electronic and phononic energy spectra. We analyze the current response to the interplay between quantum confinement effects of the electron and phonon systems in a gate-defined double quantum dot in a semiconductor nanowire. We show that current spectroscopy of inelastic transitions between the two quantum dots can be used as an experimental probe of the confined phonon environment. The resulting discrete peak structure in the measurements is explained by theoretical modeling of the confined phonon mode spectrum, where the piezoelectric coupling is of crucial importance. © 2010 The American Physical Society.