When an email template changes you also need to update all parsing rules. Most of the time, email templates do not change substantially and a quick correction will make the parsing rules work again.
Once you save the parsing rule again, the last 300 emails are re-scheduled for parsing. If your emails got parsed again with the updated parsing rules, you should see the correct results if you open one of the emails.
If you want to push this newly parsed data to another software with webhooks, you need to re-schedule them for dispatching. What you need to do is to select the last emails in the list view and choose "Re-Schedule Webhook Dispatching" in the global actions top-right.