In this article we are interested in the controllability with one single control force of parabolic systems with space-dependent zero-order coupling terms. We particularly want to emphasize that, surprisingly enough for parabolic problems, the geometry of the control domain can have an important influence on the controllability properties of the system, depending on the structure of the coupling terms. Our analysis is mainly based on a criterion given by Fattorini that reduces the problem to the study of a unique continuation property for elliptic systems. We provide several detailed examples of controllable and non-controllable systems.