The challenge of simultaneously simulating clouds and planetary-scale winds has been a key reason for uncertainty in future climate predictions, and it is unlikely that we will routinely run ensembles of storm-resolving, long-term projections before 2050. However, machine learning (ML) algorithms trained on storm-resolving models run for shorter periods can mimic the statistical effects of fine-scale clouds in less expensive climate models, thereby accelerating climate modeling. Despite this progress, statistical algorithms alone have limitations preventing their widespread adoption by the climate community. In this presentation, we will unveil recent strategies that improve the trustworthiness of data-driven parameterizations, from distilling a neural network into an interpretable nine-parameter equation to targeting climate-invariant mappings, improving neural networks' generalization in a changing climate. We will discuss these advancements in the context of cloud cover and convection parameterizations for the ICON and CAM models. These strategies can also be adapted to other parameterizations (e.g., microphysics, aerosol activation), and Earth system models like EC-Earth, which are relevant to the CleanCloud community. By making data-driven parameterizations more robust and physically consistent, we aim to develop hybrid physics-ML Earth system models that have the potential to overcome long-standing parameterization challenges and make high-quality climate models more accessible to the broader community.