This paper investigates some properties of Parallel Sequential Array Acceptors with Restricted Neighborhood (RPSA's). An RPSA is a Parallel Sequential Array Acceptor (PSA) with the restriction that the neighborhood template of each cell except the first cell is {0,1}. This paper first investigates the relationships of accepting powers between RPSA's and PSA's, and also investigates the difference between the accepting powers of one-way and two-way RPSA's, and the difference between the accepting powers of deterministic and nondeterministic RPSA's. This paper then investigates closure properties of the classes of sets accepted by one-way RPSA's under several operations.