Being an Atlassian Administrator and developer for 10 years, I've done a lot of JIRA deployments, administration, managing and developing workflows.
As an admin, You probably would know that Subtasks, actually defined as an issue bing a child of another issue (we will call it parent and child) - is an option You can select to disable in You JIRA.
It can be disabled under Gear → Issues → Sub-tasks.
Lets look at some of the things around Subtasks.
- Its an unfortunate name, as there also is a default issuetype called Sub-task (with a hyphen). This is very confusing for many users and admins, that Subtasks is a concept for at relationship, as well as an issue type called Sub-tasks. The concept could have been called child-tasks or children (to resemple the Confluence page relations)
- Subtasks always are in the same project as the parent.
- Subtasks to subtasks are not possible, there is only one sub-level
- Subtasks are not useable for Jira Service Desk Request Types.
- Jira has little awareness regarding the interaction between
Subtasks are in general fine to use, but if I were to start a new Jira installation, I might tend to start be disabling Subtasks totally.
And use Issue linking instead to have the same (and wider) possibility
Why?
Remove Confusion
You remove the confusion between the concept and the issue-type.
Tidy up the UI
Subtasks and issue links are represented in 2 different styles / ways in the UI, it does not look well, and as a user You are never sure to look for related tasks under Subtasks or Issue Links
I have grown very fond of the Issue Matrix add-on to display linked issues and issues that can be corrolated by fields.
Its awesome and flexible - and includes inline edit 
Added flexability
As Subtasks only can leve in the same project as the parent, it can be moved to other projects, this is quite a limition for many companies that has different JIRA projects for different departments/functions. Some users wont make a subtask, but a linked issue anyway.
Double programming of functionality
As Jira has no builtin awareness/functionality that relates to the part-child or linked issues, I often are asked by customers to program such things - this is often:
- Some awareness of issues blocking each other; like when a issue/subtasks is closed, other issues are made aware of this by transition or comment
- Duplication of comments between issues; some issue are added an attachment, and parent/child/linked issues are automatiocally added the same attachment
- Duplication of attachments between issues; some issue are added an attachment, and parent/child/linked issues are automatiocally added the same attachment
Very often, these functionalities need to be programmed twice; one for subtasks and one for linked issues.
Several add-ons can perform part or all of the points above from the UI - my primary favorites are JSU Automatiuon Suite and Automation.
But I mostly use JIRA Scriptrunner - I does require programming, but extends flexability of the function; the downside is its more time consuming and you have to maintain the groovy code Yourself (between upgrades etc)