Intersection of an ellipsoid and plane in parametric form

I want to find the parametric equation of the ellipse in 3d space which is formed by the intersection of a known ellipsoid and a known plane. The ellipsoid has the Cartesian equation: $(x/a)^2+(y/b)^2+(z/a)^2=1$. While the plane has the equation: $mx+ny+kz=0$. I have substituted one equation in the other but what I get is an elliptical cylinder since I had eliminated the $z$-components. How can I get the exact equation of the ellipse of intersection in parametric form?