An angular analysis of B0→J/ψK*0 and B0s→J/ψφ has been used to determine the decay amplitudes with parity-even longitudinal ( A0) and transverse ( A∥) polarization and parity-odd transverse ( A⊥) polarization. The measurements are based on 190 B0 and 40 B0s candidates obtained from 89pb−1 of p¯p collisions at the Fermilab Tevatron. The longitudinal decay amplitude dominates with |A0|2=0.59±0.06±0.01 for B0 and |A0|2=0.61±0.14±0.02 for B0s decays. The parity-odd amplitude is found to be small with |A⊥|2=0.13+0.12−0.09±0.06 for B0 and |A⊥|2=0.23±0.19±0.04 for B0s decays.