In this article, the finite-dimensional control of distributed-parameter systems with boundary control and unbounded observation is considered. This problem is solved by extending the concept of dual observer-based compensators to infinite dimensions. These compensators have the advantage that, on the one hand, the separation principle can be used to directly determine a finite-dimensional compensator and, on the other hand they can be directly applied to boundary control systems with a time-derivative of the input. In order to achieve a low controller order as well as a desired control performance, a parametric approach is proposed to parameterise the degrees of freedom contained in the controller. Consequently, desired design specifications can be achieved by using a parameter optimisation. The proposed design of finite-dimensional dual observer-based compensators is demonstrated by means of a heat-conducting system with Neumann boundary control and boundary measurements.