Yes--ellipses and parabolas and other good things can be drawn with a
straightedge and compass.
(1) use the focus-directrix definition of the conics. An old analytic
geometry book that I have calls it BOSCOVICH'S DEFINITION OF A CONIC
SECTION. "A CONIC SECTION IS A CURVE, THE DISTANCE OF ANY POINT IN WHICH
FROM A GIVEN POINT, IS TO ITS DISTANCE FROM A GIVEN STRAIGHT LINE, IN A
GIVEN RATIO. IF THE DISTANCE TO THE POINT IS EQUAL TO THE DISTANCE TO
THE LINE, THE LOCUS IS A PARABOLA; IF LESS, AN ELLIPSE; IF GREATER, AN
HYPEBOLA. IF THE DISTANCE TO THE LINE IS INFINITE, THE LOCUS IS A
CIRCLE; BUT IF THE DISTANCE TO THE POINT IS INFINITE, THE LOCUS IS A
STRAIGHT LINE."
(2) Use a MIRA--the sophisticated straight edge that reflects points and
lines. If you are not familiar with this, then check in catalogues from
such firms as DALE SEYMOUR PUBLICATIONS or CREATIVE PUBICATIONS.
Good luck!
Barnabas