Five single crystals of parahydrogen were grown near the triple point, and their Brillouin spectra examined in known crystal directions. An analysis of the measured frequency shifts of longitudinal and transverse components led to the direct determination of the adiabatic elastic constants at 13.2 K: C11 = 3.34 ± 0.05, C13 = 0.56 ± 0.03, C33 = 4.08 ± 0.06, C44 = 1.04 ± 0.03 kbar. The fifth elastic constant C12 = 1.30 ± 0.05 kbar was evaluated from the relation C11 + C12 = C13 + C33. Values of the following constants were calculated: elastic anisotropy, A = 1.02 ± 0.06, adiabatic bulk modulus, Bs = 1.73 ± 0.04 kbar, and Debye temperature, θD = 111.2 K. All of these values were compared with earlier experimental measurements. The elastic constants were found to be in good agreement with a recent self-consistent phonon calculation by Goldman.