Recently we presented a modulation-domain multichannel Kalman filtering (MKF) algorithm for speech enhancement, which jointly exploits the inter-frame modulation-domain temporal evolution of speech and the inter-channel spatial correlation to estimate the clean speech signal. The goal of speech enhancement is to suppress noise while keeping the speech undistorted, and a key problem is to achieve the best trade-off between speech distortion and noise reduction. In this paper, we extend the MKF by presenting a modulation-domain parametric MKF (PMKF) which includes a parameter that enables flexible control of the speech enhancement behaviour in each time-frequency (TF) bin. Based on the decomposition of the MKF cost function, a new cost function for PMKF is proposed, which uses the controlling parameter to weight the noise reduction and speech distortion terms. An optimal PMKF gain is derived using a minimum mean squared error (MMSE) criterion. We analyse the performance of the proposed MKF, and show its relationship to the speech distortion weighted multichannel Wiener filter (SDW-MWF). To evaluate the impact of the controlling parameter on speech enhancement performance, we further propose PMKF speech enhancement systems in which the controlling parameter is adaptively chosen in each TF bin. Experiments on a publicly available head-related impulse response (HRIR) database in different noisy and reverberant conditions demonstrate the effectiveness of the proposed method.