You can show that EA and CM are equal. You know BC and AD are equal and that A=C. That gets you BMC and EAD being congruent and thus BM=ED.

There's a proof that if one pair of opposite sides of a quadrilateral is both parallel (BE and MD are parallel because ABCD is a parallelogram) and congruent, (just proved), that the quadrilateral is a parallelogram here at the bottom of the page.