  Every Latin square has three attributes that can be even or odd, but any two
of these attributes determines the third. Hence the parity of a Latin square
has an information content of 2 bits. We extend the definition of parity from
Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the
corresponding orthogonal arrays (OA). Suppose the parity of an
$\mathrm{OA}(k,n)$ has an information content of $\dim(k,n)$ bits. We show that
$\dim(k,n) \leq {k \choose 2}-1$. For the case corresponding to projective
planes we prove a tighter bound, namely $\dim(n+1,n) \leq {n \choose 2}$ when
$n$ is odd and $\dim(n+1,n) \leq {n \choose 2}-1$ when $n$ is even. Using the
existence of MOLS with subMOLS, we prove that if $\dim(k,n)={k \choose 2}-1$
then $\dim(k,N) = {k \choose 2}-1$ for all sufficiently large $N$.
  Let the ensemble of an $\mathrm{OA}$ be the set of Latin squares derived by
interpreting any three columns of the OA as a Latin square. We demonstrate many
restrictions on the number of Latin squares of each parity that the ensemble of
an $\mathrm{OA}(k,n)$ can contain. These restrictions depend on $n\mod4$ and
give some insight as to why it is harder to build projective planes of order $n
\not= 2\mod4$ than for $n \not= 2\mod4$. For example, we prove that when $n
\not= 2\mod 4$ it is impossible to build an $\mathrm{OA}(n+1,n)$ for which all
Latin squares in the ensemble are isotopic (equivalent to each other up to
permutation of the rows, columns and symbols).
