Wait, why do you say GPUs support parametric curves? GPUs are based on triangles mostly.
You can builds all kinds on top of pixels and triangles.
Loop-Blinn, similarly, is mostly a CPU-side approach and has a lot of drawbacks, but at it's core it's using the pixel shader to define a curve profile.
Triangles aren't necessarily involved in in rendering either, see eg how the stuff on shadertoy.com works.
Again, the vast majority of use cases for GPUs is 3d vertex graphics. But they're capable of more than that. Modern GPUs are very different from early GPUs that only worked with triangles. Some of the early ones were actually ASICs, and couldn't even load different shader programs.
What you are saying is outdated by maybe a couple of decades.