Among the few examples of embedded minimal surfaces discovered before 1980 was the genus zero doubly periodic minimal surface discovered by Scherk in 1835. Since 1980, a few examples of genus one, two, and three have been discovered. In this thesis, we construct three-dimensional families of embedded doubly periodic minimal surfaces with parallel ends of arbitrarily large genus in the quotient. Each of these families limits in a foliation of [special characters omitted] by parallel planes with periodically placed points on successive planes identified. Near this limit, the surfaces look like parallel planes connected by periodically placed catenoid-shaped necks, and in the limit these necks become nodes. Thus, we consider the limit surface a noded Riemann surface. The existence of these surfaces is proven by applying Traizet's desingularization technique. We start with a finite collection of punctured spheres, with the location of the punctures satisfying a set of balance equations. Then, we construct a family of Riemann surfaces in a neighborhood of this limit surface. Finally, we construct Weierstrass data on each member of this family and solve the period problem using the implicit function theorem.