An angular analysis of B-0 --> J/psi K*(0) and B-s(0) --> J/psi phi has been used to determine the decay amplitudes with parity-even longitudinal (A(0)) and transverse (A(\ \)) polarization and parity-odd transverse (A1(-)) polarization. The measurements are based on 190 B-0 and 40 B-s(0) candidates obtained from 89 pb(-1) of (p) over barp collisions at the Fermilab Tevatron. The longitudinal decay amplitude dominates with \A(0)\(2) = 0.59 +/- 0.06 +/- 0.01 for B-0 and \A(0)\ (2) = 0.61 +/- 0.14 +/- 0.02 for B-s(0) decays. The parity-odd amplitude is found to be small with \A1(-)\(2) = 0.13(-0.09)(+0.12) +/- 0.06 for B-0 and \A1(-)\ = 0.23 +/- 0.19 +/- 0.04 for B-s(0) decays.