Precise localization in wireless sensor networks depends on distributed algorithms running on a large number of wireless nodes with reduced energy, processing and memory resources. Prerequisite for the estimation of unique nodes locations is rigid network graph. To satisfy this prerequisite network graph needs to be well connected. On the other hand, execution of distributed algorithm in the graphs with large number of edges can present significant impact on scarce resources of wireless nodes. In order to reduce number of edges in the network graph, novel distributed algorithm for network reduction is proposed. Main objective of the proposed algorithm is to remove as many edges as possible maintaining graph rigidity property. In this paper a special case of graph rigidity is considered, namely the parallel rigidity.