Reaction-diffusion systems provide a versatile framework for modeling spatially distributed biochemical dynamics. They are widely used to describe cellular processes in which spatial organization plays a key functional role, including intracellular signaling pathways and membrane-associated pattern formation. Specific examples include the spatial regulation of GTPase activity and cellular behaviors such as chemotaxis. In this thesis, reaction-diffusion systems are formulated as high-dimensional continuous-time Hidden Markov Models (HMMs), where the latent state evolves according to stochastic reaction and diffusion dynamics and observations are partial and noisy, typically involving spatially averaged measurements. Knowing the reaction and diffusion parameters is essential, as their values determine the phase of the system within its phase diagram and thus the qualitative dynamical regime.