When I get an email that someone added me as a reviewer on a pull request, I open the link and I'm in BitBucket. But there's not a good way to look at the JIRA ticket that is being referenced by the Pull Request unless I go and search for it in our JIRA board.

There should be a link on the Pull Request that allows you to link to the task in the JIRA board, especially because JIRA makes a link for the PR on their end.