I realized that there isn't currently a pre-defined checker for parametric planes (though there is one for parametric lines). So I've put together a macro file for a new ParametricPlane object that works like the ParametricLine object. I've attached the macro file for it. There are comments at the top, but it should allow you to ask questions about parametric planes similar to those about parametric lines in the problems I mentioned.
Currently, this file only works with planes in 3-space (because it uses a cross product internally). I haven't taken time to try to work out a more general comparison mechanism, but I hope that is sufficient for your needs.