Cooperative adaptive cruise control (CACC), an extension of adaptive cruise control (ACC), is an important intelligent transportation solution for future mobility. It leverages vehicle-to-vehicle information to improve longitudinal tracking, traffic throughput, energy consumption, and string stability. However, controller tuning remains sensitive to model uncertainties and communication delay. This paper presents an analytical parameter-space-based fractional-order PD (FOPD) controller tuning framework for the CACC problem. For the constant-time headway spacing policy, the fractional-order controller parameters are explored over the (kp, kd, µ) parameter space, accounting for plant uncertainties. To enable a tractable stability assessment for the commensurate fractional-order characteristic equation, a variable transformation is used to obtain an equivalent polynomial form, and stability is then verified using the Hurwitz criterion. The resulting parameter-space maps provide a transparent graphical approach to controller gain and fractional-order selection. Moreover, the CACC design is implemented by a feedforward controller that uses preceding vehicle acceleration information within the predecessor-vehicle-following communication topology. The proposed method is tested in a vehicle platoon simulation environment with respect to string stability and the time-domain responses of position, velocity, acceleration, and headway time. The fractional-order CACC is compared with integer-order controllers. Simulation studies under representative CACC maneuvers demonstrate that the proposed parameter-space approach enables systematic FOPD tuning and achieves improved performance requirements compared to integer-order PD control.