While searching for issues using the /rest/api/3/search endpoint, we can use the fields parameter to select which issue fields to include in the results.
For subtasks, the parent field can be used here and the parent field is actually a complex JSON object that includes the id, key, summary, status, priority, and issuetype of the parent.
How can I include more fields in the parent field JSON? I am particularly interested in seeing the Epic Link field value of the parent issue.
The alternative would be to collect the parent keys and make another REST request for the parent issues, which is something I'd rather not do.