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I'm no math whiz, but the actual parabola's I've seen were direct opposites of the parabolic rods I've seen--parabolas curve more toward the middle and parabolic rods flex more in the butt, with a stiffer mid-section, and a bit of flex in the tip. Semi-parabolic usually means there is a bit more flex in the tip at short distances, then the butt flexes at longer distances, supposedly to get around the parabolic taper's poor performance at short distances (also easier on fine tippets when fish take). These are tapers that are easy to make in bamboo, but harder to do in glass and graphite (due to the mechanical limitations of wrapping cloth on a mandrel), so parabolic generally means heavier tip/softer butt in those materials. To further confuse matters, the terms parabolic and semi-parabolic have been over-used in marketing to the extent that you can't rely on any maker's description, unless you've cast the rod yourself or have the word of someone you trust that it is, in fact, parabolic. Charles Ritz designed rods that had stiffer tips and softer butts for his "Long lift/Long flex" rods (like the Bi-Power and the PPP), and Pezon et Michel called them "Parabolic" when they marketed them for the public, so the name became associated with thicker tips and thinner butts from then on. Scott made bass bugging rods that used, eg, an 8wt tip on a 7wt butt. They never called them parabolic, but they sure could throw big bugs in a straight line (the tip remaining in a more level plane through the casting stroke).
-CC

At one time, Scott offered those rods with two tips (like the Orvis "Pacechanger" bamboo, but I don't recall what Scott called them, if anything). One tip would be a match to the butt (a six wt, maybe) and the other would be heavier, but with the same ferrule (7 wt), the idea being one rod for trout and bass. They brought them back a couple of years ago, but didn't keep them in the line up. I suppose tastes in rods have changed, or people who cast bass bugs regularly have found another answer (glass, maybe, or smaller bugs). I think you can still special order them through Harry Murray's fly shop (or he still has NOS rods he hasn't unloaded). Typically, parabolic rods in glass and graphite have been made of blanks from different mandrels, but it can be done on one mandrel with custom cut cloth. I'm sure Mark Steffen has done this, for Dennis Franke (Glastech) and for his own rods. Mike McFarland offers some parabolic glass that may be made on one mandrel, as well.
-CC

Convex taper is another animal, but there are some examples of each that have similarities.
-CC

The Superfines (most of them, anyway) are progressive, but slowed to the point that they are full-flexing--really more of a shallow straight taper than typical progressive. Parabolic tapers are stiffer in the middle and softer in the butt, so the upper 2/3 of the rod remains relatively straighter during the cast. The execution of that idea by various makers is where it gets complicated.

For what it's worth, Keith, I agree with your sentiment. I like experimenting with tapers, but describing them without graphics and actual rods to cast starts to get tiresome. In the end, very fine rods have been made with many different iterations of these tapers. I like casting good fly rods, period, especially when matched to the right fly line weight and taper, which is whole 'nother discussion.
-CC