  Using Traizet's regeneration method, we prove that for each positive integer
n there is a family of embedded, doubly periodic minimal surfaces with parallel
ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal
group of translations. The genus 2n-1 family converges smoothly to 2n copies of
Scherk's doubly periodic minimal surface. The only previously known doubly
periodic minimal surfaces with parallel ends and genus greater than 3 limit in
a foliation of Euclidean space by parallel planes.
