Your best option would be to use Docker and pass the Jira Software version as a parameter. You can create a base Docker image which contains installation of Jira and perhaps a default JIRA_HOME folder (if it is not to big). If you use Docker-compose you can spin up an accompanying mysql database with a prepared database.
At it’s core, the principle of Docker is to create an immutable environment. So every environment you spin up will be in the exact state as defined in the Dockerfile. As such, you can easily run your tests automatically against a multiple instances simultaneously knowing that each environment is exactly the same.
Either way, you will need the remote agent(s) to start the Docker machines. I believe there are built-in tasks in Bamboo that can take care of running Docker containers, however, I would consider it easier to just create a script that loops over an array of version number to start the tests