Heat pump and thermal energy storage (HPTES) systems, which are widely utilized in modern buildings for providing domestic hot water, contribute to a large share of household electricity consumption. With the increasing integration of renewable energy sources (RES) into modern power grids, demand-side management (DSM) becomes crucial for balancing power generation and consumption by adjusting end users' power consumption. This paper explores an energy flexible Model Predictive Control (MPC) design for a class of HPTES systems to facilitate demand-side management. The proposed DSM strategy comprises two key components: i) flexibility assessment, and ii) flexibility exploitation. Firstly, for flexibility assessment, a tailored MPC formulation, supplemented by a set of auxiliary linear constraints, is developed to quantitatively assess the flexibility potential inherent in HPTES systems. Subsequently, in flexibility exploitation, the energy flexibility is effectively harnessed in response to feasible demand response (DR) requests, which can be formulated as a standard mixed-integer MPC problem. Numerical experiments, based on a real-world HPTES installation, are conducted to demonstrate the efficacy of the proposed design.