That comes, of course, from the property of the foci of an ellipse, that the total distance from a point on the ellipse to the two foci is constant. Of course, the fact that your string has fixed length keeps the sum of those distance (the string itself) constant.
A parabola only has one focus but there is also a line (the "directrix") such that the total distance from a point on the parabola to the focus and directrix is constant. The distance from a point to a line is measured PERPENDICULAR to the line so you would have to design some mechanism to insure that your string stayed perpendicular to the line. Perhaps use a toy railroad track to represent the directrix and have the string tied to a rail truck that is free to move along the rails.
A hyperbola has two foci, like an ellipse, but it is the DIFFERENCE between the two distances, from point on the hyperbola to the two foci, that is constant. I just can't think of a mechanism that will keep the difference of the two distances constant.