It is well known that the presence of symmetry in the equations of a dynamical system has a profound effect on the resulting behavior. This note examines how this effect is manifested in the corresponding parameter identification problem. Our work shows that standard ideas such as persistent excitation in a trajectory can be explained by symmetry. Moreover, by understanding how symmetry affects the dynamics, it may be possible to obtain sufficient information to achieve full identification even when typical trajectories are not persistently exciting. Alternately, our analysis shows how properly interpreting the output of the identification process can give useful information even if full identification is not possible.