We deal with m-component vector-valued solutions to the Cauchy problem for a linear both homogeneous and nonhomogeneous weakly coupled second-order parabolic system in the layer (Formula presented.). We assume that coefficients of the system are real and depending only on t, (Formula presented.) and (Formula presented.). The homogeneous system is considered with initial data in (Formula presented.), (Formula presented.). For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to (Formula presented.), (Formula presented.). Explicit formulas for the sharp coefficients in pointwise estimates for solutions to these problems and their directional derivative are obtained. © 2020 Informa UK Limited, trading as Taylor & Francis Group.