Years ago, (1967)a fellow eccentric and I were discussing stirling engines(he was doing most of the talking and I was looking for flaws in his logic) and he made a point that I've remembered but not verified. He said that, all other things being equal, the power from the engine would be proportional to the diameter of the piston. Therefore if you double the diam, you get x4 the output. So he thought that the way to go would be enourmous pistons, probably made out of foam, with only a relatively small and slow movement. It might be possible to think in terms of the flexing of the surface of the "piston" as the movement, rather than a sliding piston in a cylnder.
We thought of a surface made up of a honeycomb of foam, each hole being filled with coarse steel wool as a "displacer" if that's the right term.
Re parabolic mirrors - start with a large hoop(plastic tube pulled into a circle) with each surface covered in aluminised melinex film and pump out the air between, each surface is a pretty good parabola. Use the pipe as an air duct as well as being the frame.
If you control the pressure, you can control the focal length.