Multitask diffusion strategies are useful to estimate node-specific, or, multiple parameter vectors over a distributed network by exploiting inter-cluster and intra-cluster cooperation. During cooperation, all nodes transmit their intermediate estimates to their neighboring nodes, resulting in high energy consumption. In this paper, we propose a clustered multitask diffusion affine projection algorithm by transmitting only a subset of the entries of the intermediate estimate vectors among the neighboring nodes. The proposed algorithm, namely, clustered multitask partial diffusion affine projection algorithm provides a trade-off between the estimation performance and the required communication cost. Important results on convergence (in mean and mean square) of the proposed strategy are presented. Numerical simulations reveal that even though the estimation performance deteriorates somewhat as the number of coefficients transmitted to the neighboring nodes decreases, the degradation can be compensated to a large extent by a proportional increase in the magnitude of the regularization strength among the clusters.