  This paper deals with two separate but related results. First we consider
weak solutions to a parabolic operator with H\"ormander vector fields. Adapting
the iteration scheme of J\"urgen Moser for elliptic and parabolic equations in
$\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the
Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i
(a_{ij} X_j u)$ we use this to show H\"older continuity. We assume the
coefficients are bounded and elliptic. The iteration scheme is a tool that may
be adapted to many settings and we extend this to nonlinear parabolic equations
of the form $u_t = -X_i^* A_j(X_j u)$. With this we show both a Harnack
inequality and H\"older continuity of weak solutions.
