  We develop weak and strong maximum principles for boundary-degenerate,
linear, parabolic, second-order partial differential operators, $Lu :=
-u_t-\tr(aD^2u)-\langle b, Du\rangle + cu$, with \emph{partial} Dirichlet
boundary conditions. The coefficient, $a(t,x)$, is assumed to vanish along a
non-empty open subset, $\mydirac_0!\sQ$, called the \emph{degenerate boundary
portion}, of the parabolic boundary, $\mydirac!\sQ$, of the domain
$\sQ\subset\RR^{d+1}$, while $a(t,x)$ may be non-zero at points in the
\emph{non-degenerate boundary portion}, $\mydirac_1!\sQ :=
\mydirac!\sQ\less\bar{\mydirac_0!\sQ}$. Points in $\mydirac_0!\sQ$ play the
same role as those in the interior of the domain, $\sQ$, and only the
non-degenerate boundary portion, $\mydirac_1!\sQ$, is required for boundary
comparisons. We also develop comparison principles and a priori maximum
principle estimates for solutions to boundary value and obstacle problems
defined by boundary-degenerate parabolic operators, again where only the
non-degenerate boundary portion, $\mydirac_1!\sQ$, is required for boundary
comparisons. Our results complement those in our previous articles
[arXiv1204.6613, arXiv:1305.5098].
