How to open new Finder tab without showing parent directory structure on the left?

What you may require is to 'root' the folder that you need into the sidebar.

Is it as simple as dragging the folder that you want into the sidebar? If not I may have misunderstood your question.

There is probably a way to default this behaviour by using default write com.apple.finder.plist <somevariable> -bool true (or similar) but I would not be able to tell you the particular variable without spending a deal of time on it. Apple prefer you to use the GUI in a lot of cases, but editing the plist files generally grant you more granular control over the system.