  We give an explicit formula for the Hilbert-Poincar\'{e} series of the parity
binomial edge ideal of a complete graph $K_{n}$ or equivalently for the ideal
generated by all $2\times 2$-permanents of a $2\times n$-matrix. It follows
that the depth and Castelnuovo-Mumford regularity of these ideals are
independent of $n$.
