We present a calculation for the beam-vector-deuteron double spin asymmetries $T^e_{10}$ and $T^e_{11}$ in elastic electron-deuteron (e-d) scattering as functions of the four-momentum transfer square $Q^2$ and the electron scattering angle in the laboratory frame $\theta_e$. Our formalism, based on the one-photon-exchange Born approximation, uses the realistic and high-precision Argonne v18 NN potential for the deuteron wave function and the standard dipole fit for the free nucleon form factors. We also study the sensitivity of our results to the D-wave component of the deuteron wave function. We find large sensitivity of the results for $T^e_{10}$ and $T^e_{11}$ to the D-wave component of the deuteron wave function at $Q^2 > 0.6\,(\text{GeV}/c)^2$.