I never used it therefore I can guess only.
I hope somebody from GS will state it more definitely.
I think a parent element is the cutted element which contains gables array (handle) in the API_ElementMemo structure ( wall, column, object etc ).
And the plane equation coefficients are relative the local coordinate system of this element.
Otherwise I think it would be required to recalculate coefficients at every moving or rotation of the element.