To prevent the parted-off piece from falling to the ways and potentially getting damaged or doing damage, mount a catcher of some sort in the tailstock so that the loose piece can't fall. As an example, if the step before parting off was to drill a hole through the piece being parted, back the drill bit out just far enough that the parting blade won't touch it and leave it there. It will catch the piece being parted off. If the bore of the parted off piece is important, replace the bit with a piece of solid or hollow round stock. Etc.