How would I build a shell of an elliptic paraboloid as accurately as possible (in terms of its mathematical properties (all parallel rays end up in its focal point)).
Preferably it should be:
Depending on the accuracy you require for your shape, you can certainly use cardboard to create the reflector. If you don't require it to be a reflector, it gets easier!
Consider that the shape is a three dimensional revolution of a two dimensional curve. One will have to presume that you can create the curve on cardboard using the formula of your choice.
Cut the curve from cardboard, then slice it in half vertically, removing a bit more from the center vertical cut line. With a sufficient number of these half-parabolas, glue them standing vertically on a suitable base (cardboard!) with all the center cuts pointing inward. The reason you remove a bit at the center cut line is that the thickness of the cardboard would otherwise prevent optimum alignment of each curved piece.
If, for example, you chose to use eight pieces of 6 mm cardboard, you'd have an octagon at the center with each face/edge of the octagon being 6 mm long. As you increase the number of "fins," the polygon in the center becomes larger and has to have that many more faces.
You can also increase the number of fins by cutting more away from the center cut and expanding the placement of that cut outward from the center.
When you have the skeleton/backbone/structure created, you can apply a surface treatment to the cardboard. This could be as simple as paper sheets glued to the cardboard edges. If a reflector is your goal, mylar film carefully stretched to the fins may provide the surface needed.
You could create the surface treatment by gluing cardboard between the fins, keeping it below the level of the cuts. Apply a layer of waterproofing such as paint or epoxy or similar and then slather on a light coat of wallboard compound. Once dry, you can sand a smooth shape to the surface to get even better results. I'm not sure how you'd get an optically reflective surface on wallboard compound, but it would provide a smoother surface for a mylar covering.
Because one word is worth one one-thousandth of a picture, I'm hopeful the attached image will be useful:
EDIT: Using a thinner material will allow more "fins" with smaller gaps and reduce the interpolation referenced in the comment section. Perhaps a primary support of cardboard with intermediate segments constructed of heavy paper with interspersed segments of ordinary paper will provide higher resolution with the compromise of being more difficult to construct!
Yet another solution, perhaps easier to construct. Consider to rotate the planes of cardboard from parallel to the axis of rotation to perpendicular to it. Solve the equation for x and make a series of circles in cardboard based on the y-thickness of the material. Thicker cardboard means lower resolution, fewer sheets to cut, thinner means higher resolution, even to the level of paper thickness.
The ease of construction comes into play because you need only to stack the sheets. You can make a negative or a positive form by using the holes or the cut-out portions. Alignment can be accomplished with a common location (the center of the cut-out circles) or a pair of holes outside of the cut-out circles, then pushing dowels through them.
I came up with another solution.
I think this should make it pretty accurate (shape-wise).
(*) The only thing I am still thinking about is out of which material would be the grey block It can't be out of clay for instance since the cardboard would not be rigid enough.
You can get a pretty consistent shape with a balloon. I feel a little silly suggesting it, but perhaps a large balloon and paper machete? The convex side would be fairly lumpy, but the concave side would be pretty smooth. You would be limited in the exact shape, though they would all be roughly parabolic. You could modify the shape somewhat by partially inflating the balloon and squeezing the side opposite the paper machete.
