"relieve the heel to save time when touching up the cutting edge"
I guess I've never felt the need to save time when sharping a parting tool. Set the platform angle, grind a couple of seconds on a 600 grit CBN, hone off any detectable burr by touching too a leather wheel, done.
FWIW, I've taken to a different grind on some of my "diamond" parting tools, a different angle on the "top" with the point meeting the widest part of the diamond cross section.
The three at the bottom show this grind.

These are the old Craftsman parting tools - the ones I use the most for general work.
I like the angle which allows entry into the wood at a slightly different tool/handle presentation which feels comfortable too me.
I find this a little easier to make a very clean cut.
It does make these tools usable with just one side up which hasn't been a problem.
The other one I use the most is the Thompson, the vertical one in this picture. The shaft on this is 5" long, about 1/2" diameter - I hold the shaft, have never put it in a handle.
I use this, and some others with flat sides, but twisting it back and forth
very slightly when making a deep sizing cut. This lets the sharp edges behind the point shave away a tiny bit of wood on both sides to increase the clearance.
I've ground the sides flat on one of the smallest tool to make the cut thinner - handy in some situations.
JKJ