We show that the problem of computing the number of perfect matchings in $K_{3,3}$-free graphs is in $NC$. This stands in striking contrast 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.