In this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form Omega = (0, pi) x Omega(2) with Omega(2) a smooth domain of RN-1, N > 1. When the control is exerted on {0} x omega(2) with omega(2) subset of Omega(2), we obtain a necessary and sufficient condition that completely characterizes the null controllability. This result is obtained through the Lebeau-Robbiano strategy and requires an upper bound of the cost of the one-dimensional boundary null control on (0, pi). The latter is obtained using the moment method and it is shown to be bounded by Ce-C/T when T goes to 0(+).