This paper is the first of two parts which together study the null controllability of a system of coupled parabolic PDEs. This work specializes to an important subclass of these control problems which are coupled by first and zero-order couplings and are, additionally, underactuated. In this paper, we pose our control problem in a fairly new framework which divides the problem into interconnected components: we refer to the first component as the analytic control problem; we refer to the second component as the algebraic control problem, where we use an algebraic method to "algebraically invert" a linear partial differential operator that describes our system. This allows us to recover null controllability by means of internal controls which appear on only a few of the equations. Treatment of the analytic control problem is deferred to the second part of this work [D. Steeves, B. Gharesifard, and A.-R. Mansouri, SIAM T. Control Optim., 57 (2019), pp. 3297-3321]. The conclusion of this two-part work is a null controllability result for the original problem.