The tracking performance of manipulators can be improved considerably by adaptive feedforward control (AFFC). However, complex kinematics hinder the application to parallel kinematic manipulators (PKMs). This paper proposes a compact and efficient formulation of the full PKM kinematics enabling real-time application of AFFC to complex PKMs. The efficient kinematic formulation is the basis for the inverse dynamics used to compute the feedforward signal. A Kalman filter is used for online estimation of the parameters in the equations of motion. A parallel multi-rate implementation is used, which, together with the efficient kinematic formulation, allows for a feedforward sampling time as low as 0.5 ms. The parameters are updated every 30ms, which suffices to track the slow parameter variations. The application to a highly repeatable flexure-based manipulator is considered. Experimental results for the manipulator show that the tracking error can be reduced by 97.5% compared to using feedback control only.