The parity conjecture for elliptic curves predicts that the rank of an elliptic curve defined over the rationals has the same parity as the pSelmer rank for a prime number p. Could anyone familiar with the recent development sketch what has happened in the last few years, and what the state of the art is?
For convenience, restrict to elliptic curves over $\mathbf{Q}$ (there are more general results/conjectures over number fields). There are three possible parities one could consider: (i) The parity of the rank of $E(\mathbf{Q})$. (ii) The parity of the $p$Selmer rank of $E$ for a prime $p$. (iii) The parity of the order of vanishing of the $L$function of $E$ (determined by the root number). Conjecturally these parities are all the same, and this follows from the BSD conjecture. The parity conjecture usually refers to the claim that (i) and (iii) are the same. The claim that (i) and (ii) are the same (your question) follows from the conjecture that Sha(E) is finite (it is almost equivalent to this conjecture), and this is only known in the cases where one can (essentially) prove the entire BSD for $E$ (for example, by work of Kolyvagin). The conjecture that (ii) and (iii) are the same is also sometimes called the parity conjecture. There has been progress on this question in recent years by Nekovar, and most recently by the brothers Dokchister. The current state of the art is that (ii) and (iii) are now known to be the same for all elliptic curves over $\mathbf{Q}$. A survey article by Tim Dokchitser can be found here: 


Let me add a few remarks to the very nice CW answer already given.


