In the properties menu you could delete all *but* the points you want to keep, then save the file.
But what I'd recommend is using the scissors to cut at the start/end point. Then switch to the merge tool, and select the piece you want. Now right click and choose "save as..." to save just that piece to a new GPX file.
Hope that makes sense. Oh, then right click on the original file in the active file list and reload it.