  This is a continuation of our combinatorial program on the enumeration of
Borel orbits in symmetric varieties of classical types. Here, we determine the
generating series the numbers of Borel orbits in
$\mathbf{SO}_{2n+1}/\mathbf{S(O}_{2p}\times \mathbf{O}_{2q+1}\mathbf{)}$ (type
$BI$) and in $\mathbf{Sp}_n/\mathbf{Sp}_p\times \mathbf{Sp}_q$ (type $CII$). In
addition, we explore relations to lattice path enumeration.
