We extend the results of the FBSDE theory in order to construct a probabilistic representation of a viscosity solution to the Cauchy problem for a system of quasilinear parabolic equations. We derive a BSDE associated with a class of quailinear parabolic system and prove the existence and uniqueness of its solution. To be able to deal with systems including nondiagonal first order terms along with the underlying diffusion process we consider its multiplicative operator functional. We essentially exploit as well the fact that the system under consideration can be reduced to a scalar equation in a enlarged phase space. This allows to obtain some comparison theorems and to prove that a solution to FBSDE gives rise to a viscosity solution of the original Cauchy problem for a system of quasilinear parabolic equations.