The Dandelin sphere of the parabola is the unique sphere, which is tangent both to the cone and the cutting plane. The point where the sphere touches the cutting plane is the focus of the parabola. The directrix is the intersection line between the cutting plane and the (orange) plane containing the (red) circle where the sphere touches the cone.
Consider a point on the parabola and the shortest segment (in orange) to the directrix. This is equal in length to the segment of generatrix pictured in purple (because the cutting plane is parallel to this generatrix), which in turn is equal to the green segment between the red and orange circle (both are segments of generatrix between the same parallel circles). Finally, the latter segment is equal to the other green segment from to the focus (since both segments are tangent to the Dandelin sphere and issue from the same point).
It follows that the distances of from the focus and the directrix are equal!