In this paper, we study the problem of local parameter identifiability for discrete dynamical systems with discrete infinite-dimensional parameter. Our results are related to the so-called conditional local parameter identifiability. In this case, one introduces a special class ${\cal P}$ of possible perturbations of the selected parameter $P^0$ and finds conditions on observations of trajectories under which all parameters $P \in {\cal P}$ close to $P^0$ coincide with $P^0$. We consider discrete dynamical systems for which the trajectories are observed at points $k=1, 2, \ldots$ or at a countable set of points $0<t_1<t_2<\ldots$. The case of a linearly perturbed diffeomorphism in a neighborhood of a hyperbolic set is also considered.