It depends; ideally, a single source of truth for the data (i.e. the third party service) is preferable in my opinion.
However, if your Requests to the third party services are slow or quota-limited, then there can be an argument for storing more of their data on your side, accepting that it might be sometimes stale. Depending on the nature of the API, and how easily updated data can be is retrieved, you can keep your version mostly up-to-date (using a scheduled job for example).