Currently, the DateInput component, used by the DatePicker, doesn't support copy/paste functionality. The pasting in particular will need to update all relevant date sections, which currently is disabled. The enhancement request was logged in the component repo, which is private. I moved the thread into the public repo in order to be able to subscribe to it.
Please note that although this functionality has a high priority, we will not be able to provide any specific time frame for its implementation. I would suggest you follow the Github thread for more details.