It seems like the problem of trying to estimate model parameters for continuous time models is not commonly covered in standard econometric textbooks, even those focusing on time series. I certainly am able to read and work on research pertaining to discrete time series models, but I have never encountered an introduction on how to estimate continuous time models using discretely sampled data and I would be curious to know if anyone has suggestions.
This is seen as a bit of a niche field, which is likely why there are not so many books and these issues are not covered in standard econometrics texts. Options pricing models are usually fitted to options data rather than estimated econometrically from historical data. For statistical models, it is often more convenient to start from a discrete model as the data is discrete anyway.
However, there are some relevant books. You could check e.g. "High-Frequency Financial Econometrics" by Ait-Sahalia and Jacod, which covers the estimation of diffusion and jump models.