Model Predictive Control (MPC) is able to directly deal with dead-time (DT) phenomena. Nevertheless, implicit delay compensation heavily affects computational aspects of such control algorithm. The corresponding stability and feasibility analyses also become numerically tougher. Recently, the Linear Parameter Varying (LPV) toolkit has become a popular representation for complex, nonlinear dynamics, with corresponding MPC algorithms being developed over the last few years. In this context, we propose a novel MPC scheme for DT LPV systems, which is able to ensure input-to-state stability, recursive feasibility, and constraint satisfaction. The method operates under a DT compensation framework and uses a DT-free model. Thereby, the DT is explicitly accounted for, and thus, augmented state-space models are avoided. The scheme also offers robustness properties, which are synthesised w.r.t. model-mismatches and delay estimation uncertainties. A solar collector benchmark example is used to illustrate the advantages of the method, which is compared against a regular LPV MPC algorithm (with standard implicit DT compensation).