No, this is not possible - in StarRC, or in any other industry standard parasitic extraction tool.
(actually, the goal as you formulated it is not well-defined - what if your net goes to other cells (at the same or lower hierarchy level)).
What you can do, to speed up your extraction, is to extract (at the top hierarchy level, of course) only the net(s) that you are interested in.
In some extraction tools this is called "selective extraction".
Of course, the LVS/DRC should process the whole design (otherwise, how can net connectivity be extracted?), to prepare the proper input data for extraction tool (CCI database, LVS database, etc.), and this now may become your overall extraction time bottleneck.
There is an alternative technique, less accurate one (but potentially much faster), is to use what's called "hierarchical extraction", i.e. to black-box (or to grey-box) the cells that you are not interested in.
Max