  In this paper, we first study the dual fractional parabolic equation
\begin{equation*} \partial^\alpha_t u(x,t)+(-\Delta)^s u(x,t) = f(u(x,t))\ \
\mbox{in}\ \ B_1(0)\times\R , \end{equation*} subject to the vanishing exterior
condition. We show that for each $t\in\R$, the positive bounded solution
$u(\cdot,t)$ must be radially symmetric and strictly decreasing about the
origin in the unit ball in $\R^n$.
  To overcome the challenges caused by the dual non-locality of the operator
$\partial^\alpha_t+(-\Delta)^s$, some novel techniques were introduced.
  Then we establish the Liouville theorem for the homogeneous equation in the
whole space \begin{equation*}\label{B} \partial^\alpha_t u(x,t)+(-\Delta)^s
u(x,t) = 0\ \ \mbox{in}\ \ \R^n\times\R . \end{equation*}
  We first prove a maximum principle in unbounded domains for anti-symmetric
functions to deduce that $u(x,t)$ must be constant with respect to $x.$
  Then it suffices for us to establish the Liouville theorem for the Marchaud
fractional equation
  \begin{equation*} \partial^\alpha_t u(t) = 0\ \ \mbox{in}\ \ \R .
\end{equation*} To circumvent the difficulties arising from the nonlocal and
one-sided nature of the operator $\partial_t^\alpha$, we bring in some new
ideas and simpler approaches. Instead of disturbing the anti-symmetric
function, we employ a perturbation technique directly on the solution $u(t)$
itself. This method provides a more concise and intuitive route to establish
the Liouville theorem for one-sided operators $\partial_t^\alpha$, including
even more general Marchaud time derivatives.
