  Aim of this paper is to extend the continuous dependence estimates proved in
\cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic
equations. As by-product of these estimates, we get an H\"older estimate for
bounded solutions of systems and a rate of convergence estimate for the
vanishing viscosity approximation. In the second part of the paper we employ
similar techniques to study the periodic homogenization of quasi-monotone
systems of fully nonlinear second-order uniformly parabolic equations. Finally,
some examples are discussed.
