  The subject of this paper are partial geometries $pg(s,t,\alpha)$ with
parameters $s=d(d'-1), \ t=d'(d-1), \ \alpha=(d-1)(d'-1)$, $d, d' \ge 2$. In
all known examples, $q=dd'$ is a power of 2 and the partial geometry arises
from a maximal arc of degree $d$ or $d'$ in a projective plane of order $q$ via
a known construction due to Thas \cite{Thas73} and Wallis \cite{W}, with a
single known exception of a partial geometry $pg(4,6,3)$ found by Mathon
\cite{Math} that is not associated with a maximal arc in the projective plane
of order 8. A parallel class of lines is a set of pairwise disjoint lines that
covers the point set. Two parallel classes are called orthogonal if they share
exactly one line. An upper bound on the maximum number of pairwise orthogonal
parallel classes in a partial geometry $G$ with parameters
$pg(d(d'-1),d'(d-1),(d-1)(d'-1))$ is proved, and it is shown that a necessary
and sufficient condition for $G$ to arise from a maximal arc of degree $d$ or
$d'$ in a projective plane of order $q=dd'$ is that both $G$ and its dual
geometry contain sets of pairwise orthogonal parallel classes that meet the
upper bound. An alternative construction of Mathon's partial geometry is
presented, and the new necessary condition is used to demonstrate why this
partial geometry is not associated with any maximal arc in the projective plane
of order 8. The partial geometries associated with all known maximal arcs in
projective planes of order 16 are classified up to isomorphism, and their
parallel classes of lines and the 2-rank of their incidence matrices are
computed. Based on these results, some open problems and conjectures are
formulated.
