Deterministic Parallel algorithm for perfect matching in general graphs?
$NC$ algorithms for perfect matchings in general graphs is still open but there has been some progress. Here are a few that I am aware of:

For general graphs, Agrawal-Hoang-Thierauf showed that given the promise that the number of perfect matchings is small, there is an $NC^2$ algorithm to enumerate all of them. 

For the class of planar graphs, the pfaffian plays a big role. Kastelyn showed how every planar graph can be oriented in a way such that the pfaffian exactly equals the number of perfect matchings. (This was used by Valiant in to give "Holographic algorithms" for various problems) Mahajan-Subramanya-Vinay showed how the pfaffian can be computed in $NC$ using modifications of clow sequences. (Kastelyn in fact gives an algorithm to find the embedding in $P$ but I'm not sure if the pfaffian embedding can also be computed in $NC$; if yes, that would mean that counting perfect matchings in planar graphs is in $NC$.)

And a recent result of Vinodchandran-Tewari show that the isolation lemma can be "derandomized" for planar graphs (using Green's theorem!) to put planar reachability in $UL$. But $NC$ algorithms for planar matchings are still open (thanks to Raghunath for correcting my claim that it is in $UL$). An $NC$ algorithm for bipartite planar matchings was given by Datta-Kulkarni-Roy

Hope this helps.