In this paper we construct upper bounds for the solutions u({\bf{x}},t) and its gradient |\nabla u| of a class of parabolic initial-boundary value problems in terms of the solution \psi ({\bf{x}}) of the {\text S}^t-Venant problem. These bounds are sharp in the sense that they coincide with the exact values of u and |\nabla u| for appropriate geometry and appropriate initial conditions.