The main concern of this article is to investigate the boundary controllability of some $2\times 2$ one-dimensional parabolic systems with both the interior and boundary couplings: the interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting on only one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.