Question

How does one go about finding plane intersection points with a set of three parametric equations (x,y,z)?

Answer 1

Solve them.

Answer 2

You mean equations like ax+by+cz+d=0?
In that case, it's basically finding simultaneous solution to the equations of the planes. You can do this by matrices, or direct substitution.

Answer 3

http://mathworld.wolfram.com/Plane-PlaneIntersection.html