From reading Parachute Recovery Systems Design Manual T. W. Knacke, 1992 - I came away with this understanding.
It is easier to build a stable round parachute than any other shape. The goal becomes finding a circular shape that maximizes drag with the least material. The answer seems to be the quarter-spherical parachute.
Cutting gores for the quarter-spherical shape is a pain. So they searched for a good approximation. For sub-sonic parachutes, the triconical shape appears to be the best practical answer.
Other design factors such as reefing, venting, suspension line length, riser length, and skirting all impact parachute performance. So it looks to me like there is no single answer to the question.
But I sure have had a lot of fun trying things out.