Yes, a truly parametric object that is generated on the fly whenever the frame changes. Yes, you can animate the inner and outer radius parameters. The object parameters show up in the Custom Properties panel in the object sub-context.
To install the AddOn into your system simply save the text document addon_parm_torus.py (which is contained within the BLEND file) into your addons folder and then activate it in the User Preferences dialog. The object shows up in the View3D > Add > Mesh SHIFT-A menu.
This code uses the twisted torus generator from Paulo Gomes’ twisted torus addon.
More than likely, you set the segments to 0 this causes the update to crash. If you keep the segments positive, the script should continue to work.
Ideally, the Custom Properties panel should be replaced with a Properties layout panel. But for the time being I wanted to get the concept of animated properties out there so others could try it out.
It works with a mesh material now. You can only have one material per torus, but the material does get transferred to the newly generated mesh every frame.
Better scene recovery if script or empty are accidentally erased from the scene. Simply add another Parametric Torus to the scene and it will be restored.
Added bounding to the values in the Custom Properties. This should prevent crashing due to invalid parameters being passed to the generator.
Added the ‘Twist’ parameter from Paulo Gomes’ original script as an additional property you can animate.
Multiple Tori (is that the plural?) are now possible. You can have more than one parametric mesh in the scene now.
These fixes are incorporated into the version 1f release available at the top of this post.
Also, hopefully, a final render bug fix on version 1h.