Depends on the shape of the model and what you want to predict.
Given that you’re fitting a point estimate, then you probably want to do prediction by taking the point estimate and plugging it into the likelihood. In general, you can do that within the generated quantities block of a Stan program and then make predictions at the same time as you fit.
Alternatively, you can extract the parameters and implement the predictive model on the outside in Python. I’m not sure if PyStan lets you extract Stan functions (RStan does)—if it does, that’d be one way to write a prediction function.
If you do it within the generated quantities block in Stan, you can always run full Bayes over the model and get uncertainties in the predictions.