As model problem, we consider a PDE-constrained optimization problem of tracking type with the heat equation as parabolic state equation. The solution is characterized by the Karush–Kuhn–Tucker (KKT) system. We are interested in a preconditioner for this system that is robust in the grid sizes, the regularization parameter and the diffusion coefficient. Several preconditioning techniques that are tailored towards robustness are restricted to the case of distributed control and distributed observation. We are interested in a preconditioner that does not have this limitation. We formulate the KKT system using a strong variational formulation of the state equation and a super weak formulation of the adjoined state equation. Using this formulation, we are able to construct such a robust preconditioner also for the case of limited observation or control. In order to discretize the problem, we use Isogeometric Analysis since it allows the construction of sufficiently smooth basis functions effortlessly. To realize the preconditioner, one has to solve a problem over the whole space time cylinder that is elliptic with respect to certain non-standard norms. Using a fast diagonalization approach in time, we reformulate the problem as a collection of elliptic problems in space only. These problems are not only smaller, but our approach also allows to solve them in a time-parallel way. We see that, alternatively, a space-time multigrid solver can be chosen to realize the preconditioner. We show the efficiency of the preconditioner by rigorous analysis and illustrate it with numerical experiments.