Summary

Currently, there is no way to find parent issues with/without any child issues in Jira, however, the opposite way is feasible: 

"parent is empty" can be used in JQL to fetch issues without a parent issue. 

Suggestion

It would be great to have the ability to search for all issues with/without child issues, this could be used in many business scenarios. 

Workaround

There's no native workaround to this problem; however, some Marketplace apps are capable of performing this search.