We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance in G of a template type T can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same module. The mapping algorithm should be fast in terms of data access, minimize the required number of memory modules, and guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates (i.e., any node of G along with all its neighbors) and we study the star-access problem on two specific host graphs, tori and hypercubes. We propose conflict-free mappings that are fast and load-balanced, using an optimal or provably good number of memory modules.