The double-$\ensuremath{\beta}$-decay $Q$ values of $^{130}\mathrm{Te}$, $^{128}\mathrm{Te}$, and $^{120}\mathrm{Te}$ have been determined from parent-daughter mass differences measured with the Canadian Penning Trap mass spectrometer. The $^{132}\mathrm{Xe}\text{\ensuremath{-}}^{129}\mathrm{Xe}$ mass difference, which is precisely known, was also determined to confirm the accuracy of these results. The $^{130}\mathrm{Te}$ $Q$ value was found to be $2527.01\ifmmode\pm\else\textpm\fi{}0.32$ keV, which is 3.3 keV lower than the 2003 Atomic Mass Evaluation recommended value and is consistent with another recent Penning trap measurement. The $^{128}\mathrm{Te}$ and $^{120}\mathrm{Te}$ $Q$ values were found to be $865.87\ifmmode\pm\else\textpm\fi{}1.31$ and $1714.81\ifmmode\pm\else\textpm\fi{}1.25$ keV, respectively. For $^{120}\mathrm{Te}$, this reduction in uncertainty of nearly a factor of 8 opens up the possibility of using this isotope for sensitive searches for neutrinoless double-electron capture and electron capture with ${\ensuremath{\beta}}^{+}$emission.