Weights/parameters are configuration settings not training data storage. When weights/neurons/parameters are updated after each training loop, you are essentially updating configuration settings that direct generations, not storing any particular training text or image.
Weights are what take up the space. The bigger the parameter size/the number of weights, the bigger the size of the model.
Image generators don't need the huge parameter numbers text generators need to be useful. What they need to learn simply isn't as complex.