Geological parameterization procedures entail the mapping of a high-dimensional geomodel to a low-dimensional latent variable. These parameterizations can be very useful for history matching because the number of variables to be calibrated is greatly reduced, and the mapping can be constructed such that geological realism is automatically preserved. In this work, a parameterization method based on generative latent diffusion models (LDMs) is developed for 3D channel-levee-mud systems. Geomodels with variable scenario parameters, specifically mud fraction, channel orientation, and channel width, are considered. A perceptual loss term is included during training to improve geological realism. For any set of scenario parameters, an (essentially) infinite number of realizations can be generated, so our LDM parameterizes over a very wide model space. New realizations constructed using the LDM procedure are shown to closely resemble reference geomodels, both visually and in terms of one- and two-point spatial statistics. Flow response distributions, for a specified set of injection and production wells, are also shown to be in close agreement between the two sets of models. The parameterization method is applied for ensemble-based history matching, with model updates performed in the LDM latent space, for cases involving geological scenario uncertainty. For three synthetic true models corresponding to different geological scenarios, we observe clear uncertainty reduction in both production forecasts and geological scenario parameters. The overall method is additionally shown to provide posterior geomodels consistent with the synthetic true model in each case.