You can make a thin parting tool from a saw blade. Alan Lacer shows how he does it in his DVD Woodturning Shopmade Tools and Jigs.
I suspect there is also plenty of info online. Basically, find a good saw blade, grind off the teeth, sharpen the end, and make a handle.
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I've also used small hand saws and japanese style pull saws, but only with the lathe off. That makes it difficult to make a straight cut.
With the wood turning I'd definitely use the reverse.
Edit: I see a lot of youtube videos on making parting tools. Remember, there are good videos and some not so good and some just bad.
JKJ