Wave energy is a significant source of renewable energy harnessed by wave energy converters (WECs). However, due to the relatively high levelised cost of energy (LCoE), wave energy has not attained a commercial stage yet. To minimise the LCoE, since the optimum (uncontrolled) WEC design typically differs from the optimum controlled WEC design, control co-design (CCD) techniques are essential. With CCD, the WEC control-related aspects are taken into account from the start of the WEC design phase and, ideally, the best control-informed WEC design is then achieved. This paper specifically focuses on CCD for an oscillating-watercolumn (OWC) WEC, equipped with a Wells turbine and a bypass valve. In essence, a parametric CCD approach is devised to find the optimum (control-informed) turbine rotor diameter, and bypass valve diameter, for the considered OWC WEC. In particular, the optimum design parameters minimise a ‘simplified’ LCoE, which is chosen as a suitable performance function. Despite the LCoE is primarily sensitive to the power take-off size, rather than to the bypass valve size, peak-shaving control with a bypass valve potentially increases the capacity factor and, consequently, can minimise the LCoE for small-to-medium sized turbines.