Geological parameterization entails the representation of a geomodel using a small set of latent variables and a mapping from these variables to grid-block properties such as porosity and permeability. Parameterization is useful for data assimilation, as it maintains geological realism while reducing the number of variables to be determined. Diffusion models are a recent class of generative deep-learning models that are trained to "denoise", which enables them to generate new geological realizations from a random noise input. Specifically, latent diffusion models combine the generative capabilities of diffusion with an autoencoder low-dimensional latent space. Diffusion-generated models can accurately reproduce complex 2D and 3D geological features, and correctly capture the variability in their distribution. Due to the smoothness of the parameterization, diffusion models can also be used for data assimilation in the latent space. Our current application involves conditional 2D three-facies (channel-levee-mud) systems. Significant uncertainty reduction, posterior P10-P90 forecasts that bracket observed data, and consistent posterior geomodels, are achieved.