The nni-distance is a well-known distance measure for phylogenetic trees. We construct an efficient parallel approximation algorithm for the nni-distance in the CRCW-PRAM model running in O(log n) time on O(n) processors. Given two phylogenetic trees T1 and T2 on the same set of taxa and with the same multi-set of edge-weights, the algorithm constructs a sequence of nni-operations of weight at most O(log n) \cdot opt, where opt denotes the minimum weight of a sequence of nni-operations transforming T1 into T2 . This algorithm is based on the sequential approximation algorithm for the nni-distance given by DasGupta et al. (2000). Furthermore, we show that the problem of identifying so called good edge-pairs between two weighted phylogenies can be computed in O(log n) time on O(n log n) processors.