The development of Digital Twins for cultural heritage assets requires computational models that are both physically consistent and efficient enough to support real-time analysis and long-term predictive maintenance. In this context, Scientific Machine Learning and Model Order Reduction emerge as key enabling technologies for bridging high-fidelity simulations and practical decision-support systems. In this work, we propose a digital framework that integrates Physics-Informed Neural Networks (PINNs) and Reduced Order Models (ROMs) to address parameter-dependent, time-evolving partial differential equations arising in the monitoring and preservation of cultural heritage. By embedding governing physical laws and boundary/initial conditions into the learning process, PINNs enable robust parameter identification from data, while ROM techniques provide computationally efficient, low-dimensional surrogates that are essential for scalable Digital Twin workflows. The work discusses methodological aspects relevant to Digital Twin deployment, including sampling strategies, generalization across parametric spaces, and reliability of hybrid physics-based and data-driven predictions in multi-scale scenarios. Simulated case studies based on time-dependent parametric PDEs are used to assess the proposed approach. The results demonstrate that integrating PINNs and ROMs into the unified framework can deliver computationally efficient and physics-consistent approximations, making them well-suited as core components of Digital Twins aimed at predictive maintenance and informed conservation strategies for cultural heritage assets.