One of the biggest open questions in combinatorics is whether or not there
exists a projective plane for non-prime power orders. There is a well-known
correspondence between finite projective planes and orthogonal arrays with the
maximum number of columns. We present a notion of parity of an orthogonal array
and discuss parity restrictions that give some insight as to why it may be
harder to build projective planes of order $n \equiv 2\mod4$, $n>2$, (which are
widely believed not to exist) compared to other orders.