Influence of the D-wave component of the deuteron wave functions on tensor spin asymmetries in the elastic electron-deuteron ($e$-$d$) scattering process with a polarized deuteron target and an unpolarized electron beam is investigated. We present numerical results for the tensor spin asymmetries $T_{20}$, $T_{21}$, and $T_{22}$ as functions of the four-momentum transfer square $Q^2$ and the electron scattering angle in the laboratory frame $ heta_e$. The analysis is based on a $e d \to e' d'$ reaction model in which the realistic and high-precision Argonne v18 NN potential is used for the deuteron wave functions and the standard dipole fit for the proton and neutron form factors is used for the free nucleon electromagnetic form factors. We found that the results for tensor spin asymmetries at $Q^2 > 0.5$ (GeV/c)$^2$ are sensitive to the D-wave component of the deuteron wave functions.