I have asked the same question 2 years ago in this post:
I am hoping someone can help with an API or script to update the status of linked test case in TestRail, either in an instant or scheduled, when the status of issue in Jira is changed. For example, if the issue in Jira is changed to status 'For Retest', the linked test case will automatically be changed to 'For Retest' as well. In the same manner, another script will do the same from TestRail to Jira.
First thing: I am not using TestRail, so my suggestion is more of a "how to look for the answer..."
Have you checked the vendor's documentation for TestRail to see if there is a REST API which could be called, perhaps from an Automation for Jira rule when the issue transitions?
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