Several people have mentioned 1/16" parting tools. These are probably the most common commercial 'thin' parting tools. I have the Ben's Best from PSI (advertised as a 'tenoning tool'), but I have modified it to look like the Henry Taylor tool that John mentioned by grinding a second bevel. Frankly, I don't know if that makes any difference in its performance.
The one thing to be aware of with these tools is that because the blade has a uniform thickness, it tends to rub the edges of the cut. And because the blade is fairly broad (1.25" on Ben's Best), there's a lot of material to bind and cause friction. This means that it has a tendency to burn the timber on a deep cut. It is possible to manage this problem in the way the tool is used - cut in a bit, withdraw the blade, and then do a relief cut to one side about half the width of the blade - but that doesn't completely eliminate the concern. A secondary consequence is that depending on the chemical content of the timber (I suspect on the sugar content in the fibers of the wood), that burning effect leaves a tar-like char on the sides of the blade. It's easy to remove using some fine sandpaper or a diamond hone, but I suspect that if you do that repeatedly, the thickness of the blade will become slightly tapered.
It is possible to make thin parting tools from hacksaw or sawsall blades, or small bits of high-speed steel, and the cost is even cheaper than buying one of the 1/16" tools ($15 for Ben's Best, $26 for Henry Taylor). However, my experience is that the problem with shop-made tools of this size is that the width of the blade is less than the fairly broad dimension of the commercial tools, and the result is that they tend to flex a bit leading to chatter and difficulty in controlling the cut. I have successfully used a tool of this sort on a thin spindle, but I find that the broader commercial tool is easier to control if the spindle is more than, say, 1/2" in diameter.