This paper is the second of two parts which together study the null controllability of a system of coupled parabolic PDEs. Our work specializes to an important subclass of these control problems which are coupled by first- and zero-order couplings and are, additionally, underactuated. In the first part of our work [D. Steeves, B. Gharesifard, and A.-R. Mansouri, SIAM T. Control Optim., 57 (2019), pp. 3272-3296], we posed our control problem in a framework which divided the problem into interconnected components: the algebraic control problem, which was the focus of the first part, and the analytic control problem, whose treatment was deferred to this paper. We use slightly nonclassical techniques to prove null controllability of the analytic control problem by means of internal controls appearing on every equation. We combine our previous results in [D. Steeves, B. Gharesifard, and A.-R. Mansouri, SIAM T. Control Optim., 57 (2019), pp. 3272-3296] with the ones derived below to establish a null controllability result for the original problem.