Will,
I would recommend against using a parting tool for ending your bowl turning. As you obviously know, it puts lateral force into the mix, and with face grain oriented wood, this is not a good match. Yes, you can do it, and lots of people do, but there's an easy way to take off the tenon without lateral stress. You need the 'nibbling cut'. (I made that up, famous people probably have a better name for it)
With the bowl reversed, you use a bowl gouge, or even a spindle gouge, with the flute roughly horizontal/closed. Press the lower tip of the gouge into the flat surface of the tenon, pressing toward the mounted rim of the bowl/headstock. You don't take a giant cut. Just press in until you are near the bottom of the bowl, nibbling a little off the edge of the tenon. Then advance and do it again, a short way closer to the center of the tenon. This puts any pressure toward the headstock/chuck, and there is no lateral pressure. This nibbling cut does not cut into the end grain, so it won't grab and it cuts easily.
Keep nibbling and you will end up with a rough or ridged surface once most of the tenon is gone, with a nib in the center due to the tailstock. You can make a pass or two with your gouge in a normal push cut fashion, taking a light cut, toward the center of the tenon to get a smooth, flat surface. If you want a curved bottom, and who doesn't, you can do the same with a light pull cut, shaping the curve as you go. (As with everything in woodturning, this is simple to understand when you see it but hard to describe--you can see it at 23+ minutes in Al's video)
Give it a try, I think you'll like it.