Check what exactly is holding your carriage down before you do upside down parting. There is precious little but a few bolts, even on a good sized lathe. If you have an accidental pileup, it could wreck something pretty bad. Big lathes have a fair amount of mass in the carriage which helps hold it down on the bed, but a little lathe carriage has not got this characteristic so the parting forces (which are dependent on material and blade width and rake angles) are the same no matter what size lathe is being used.) I've never needed any upside down parting tricks to make an inserted tool work. All I can see the upside down tool accomplishing is serving as a sort of damper for a machine that is out of adjustment. But its not the proper way to direct cutting forces into the bed of a typical lathe with plain ways.
How much slop does your spindle have? Is your chuck any good? Wallowed out 3 jaw chucks are notorious for looking like they hold something, but really, they don't hold it securely at all. Whatever slack there is in the slideways that the chuck jaws move in, translates into lifting of the jaws. This typically means that the jaws hold tight towards the back, and looser at the front. It may even be possible to slip shimstock underneath the front of the jaws when the chuck is tightened.
The cure for this is to rebore or regrind the jaws with a preload applied that tilts them outwards as if they were in use, so you can straighten up the bore of the jaws.
One way to detect how good your chuck is holding is to put a nice round piece in, fully engaging the jaws, and about 6 or 8" hanging out. Snug the chuck up as per usual torque. Put a dial indicator on the far end of the bar. Take a soft mallet and tap the side of the bar near the end. See how easily it deflects into wobble mode. This is not a centering test, this is a holding test. IF the part barely moves off no matter where you hit it, then the chuck is holding well. IF it moves .03 or .06 out of wack very easily, then the chuck jaws need the servicing prescribed above. (Lift the indicator stylus off the part before smacking it!).
Once the jaws hold firmly for their entire length on the part, then you have a hope in hell of parting without extreme vibration.
I'd throw HSS parting blades in the scrap bin, except that they work very well in brass, bronze and cast iron Get an insert tool for superior performance. But having said that, the insert tool is not going to fix the problems that it sounds as though your lathe is having.
Also make sure that the gibs on the compound are snug. It should be difficult to crank the compound back and forth. This is something you might have to adjust from time to time, tighten for parting, loosen for taper cutting with the compound rest.
A lot of cheap lathes have setscrew gib adjustments. Those are crap because they continually come loose or worm their way into the shim. You'd be better off the remove the setscrews, and place shimstock behind the gib to fill the available space and thus custom fit your gib to the dovetail. You might have to scrape the rough machined surface inside the dovetail where the setscrews poke through, so that you can get a full length of shimstock in there. (I went through this exercise when I used to have a Palmgren millling attachment on my lathe. It worked sweet after this alteration).
Then put the setscrews back in to keep the gib from slipping out of the crack.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)