Their should be a way in which we can run parallel sprints in team managed projects aswell. What could be the disadvantage of not having it?
I agree with @Philipp Barry , creating a separate board for each team is easily done via your filter for the board and gets you away from viewing multiple active sprints on a single board.
You can still set up a consolidated board if the need arises where all sprints are visible, but in my experience with connecting boards to Jira Align, the single active sprints per board is best and recommended.
This is a great article to review regarding sprint best practices with Jira Algn
https://community.atlassian.com/t5/Jira-Align-articles/Jira-and-Jira-Align-Integration-Sprint-Best-Practices/ba-p/1907564