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