We prove essential boundedness of the weak solutions to the Cauchy–Dirichlet problem for the quasilinear parabolic system \( {\textbf{u}}_t- \mathrm {div\,}\big ({\textbf{A}}(x,t,{\textbf{u}},D{\textbf{u}})\big )= {\textbf{b}}(x,t,{\textbf{u}},D{\textbf{u}}) \) which is modeled on the p-Laplacian vectorial operator. The nonlinear terms are given by Carathéodory functions and support controlled growth with respect to \({\textbf{u}}\) and \(D{\textbf{u}},\) while their dependence on (x, t) is expressed in terms of suitable Lebesgue scales. Our result is proved by assuming additionally componentwise coercivity of the system and appropriate componentwise control of the lower-order terms.
