https://confluence.atlassian.com/display/JIRA/Managing+Global+Permissions lists exclusions under the section:
"People who have the JIRA Administrators permission (and not the JIRA System Administrators permission) cannot do the following"

However, the list of exclusions does not mention plugin installations.

I have confirmed that "JIRA Administrators" permission is insufficient to upload a plugin, you require "JIRA System Administrators" permission.

I recommend adding this exclusion to the list to make it clear JIRA Admins will not have enough permission to install a plugin.