Hi arcos,
I agree with all of the reasoning and comments above, however...........I do part off with the lathe running but take various things into consideration.
The first thing is experience, now I know that old cherry about familiarity....etc etc. but as a full time turner you get to realise the risks you are taking in as much as danger to fingers/eyes and the finished product. So for the time being I would advocate stopping the lathe and cutting off with a fine toothed saw, you may think differently once you have gained more experience.
I have seen a saw used with the lathe running and have tried it myself but have not found it to be desirable because the saw becomes unwieldy and if it warms up it expands and jams in the cut. So I part off all the way leaving enough wood on the base to allow me to reverse the piece I am making removing any torn grain and clean up the bottom/apply decoration etc.
Obviously if it is a large piece, that is a piece that will not fit in my hand, I part off nearly all the way through, and then reverse it and finish off. On occasions I will remove the piece from the lathe and cut through either by hand or on the bandsaw, but always using a sled to support the work and prevent the bandsaw from grabbing.
The best advice I can give you is sharpen your parting tool before you start the cut and depending on the thickness of the piece being parted widen the cut slightly to prevent tool grab.
Above all else, before you carry out an unfamiliar operation on the lathe (or any other machine come to that)stop and think of what could go wrong and what the consequences could be.
You can always turn another bit but it's very difficult to replace fingers.
Regards
John BHT