We show that the problem of computing the number of perfect matchings in K 3,3 -free graphs is in NC. This contrasts with the #P-completeness of counting the number of perfect matchings in arbitrary graphs. As corollaries we obtain NC algorithms for checking if a given K 3,3 -free graph has a perfect matching and if it has an EXACT MATCHING. Our result also opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K 3,3 -free graphs.