We present the most precise determination to date of the ground-state masses of the triply charmed baryons with both parities, obtained by continuum extrapolation and fully addressing the systematic uncertainties. The calculations are performed on six $N_f=2+1+1$ HISQ ensembles, generated by the MILC collaboration, with two complementary setups for the valence charm action, one using the HISQ action and the other using the overlap fermion action. Our prediction for the mass of the lowest two triply charmed spin-3/2 baryons are: $M_{\Omega_{ccc}} (3/2^{+}) = 4793 (5) \left(^{+11}_{-8}\right)$ MeV, and $M_{\Omega_{ccc}} (3/2^{-}) = 5094 (12) \left(^{+19}_{-17}\right)$ MeV.