Parting is always a struggle, and speaking for myself, one should seldom use a as parted surface as a finished surface, the finish and accuracy are seldom there, so cut off and reface should be the normal practice. My best luck with parting with HSS is the Tee type HSS blades, I generally use one 1/8" wide. My early experiments with insert carbide parting was not so good, but after years of sacrifice and self denial, I am now able to accomplish it, even on large diameters without lubricant; it does not want to be babied, enough feed to make constant curled chips is necessary; too little feed makes the chips jam up in the cut and disaster is not far behind; the thing is that the chips are deformed narrower than the cut when sufficient feed is applied, and they can freely exit the cut; when insufficient feed is applied, they do not narrow up, and are likely to pile up and bind. In parting steel, I like to use TapFree as the lubricant with HSS tools, with carbide I use Cimcool soluble coolant in water. As I said previously, with the carbide parting inserts, cutting can often be done dry. If the parting tools are not sharp, especially the corners must not be dull, do not expect anything but poor results.
I appreciated the comments regarding the tapered blade type parting tool needing to be adjusted to fit the holder so that there is clearance on both sides of the tool; if one uses this type of tool, perhaps the tool holder should be altered so that the parting tool can lie against the face of the holder and be centered as he suggests. I think using the Tee type tool makes this of little importance, as there is more clearance, especially at the extreme bottom of the tool bit. I have seen parting tools in HSS that have a concave ground in the length of the top of the blade, I assume this acts to narrow the chip so that it can more freely exit the cut, much like the carbide insert tool does.