A graph 1 is parity embedded in a surface if a closed path in the graph is orientation preserving or reversing according as its length is even or odd. The parity demigenus of 1 is the minimum of 2&amp;/(S) (where / is Euler characteristic) over all surfaces S in which 1 can be parity embedded. We calculate the maximum parity demigenus over all loopless graphs of order n. As a corollary we strengthen the calculation by Jungerman, Stahl, and White of the genus of Kn, n with a perfect matching removed. We conclude by discussing numerous related problems