Let's use a polar formula for a conic curve,
cc[u_, eps_] := {1/(1 + eps Cos[u]) Cos[u], 1/(1 + eps Cos[u]) Sin[u]}
For -1 < eps < 1
we have an ellipse, for eps == 1
a parabola, and for eps > 1
a hyperbola.
We assume a simple function to describe dependence of the color on the angle:
ParametricPlot[ cc[u, 2/3], {u, 0, 2 Pi}, Axes -> False, PlotStyle -> Thick,
ColorFunction -> Function[{x, y, u}, Hue[u/(2 Pi)]],
ColorFunctionScaling -> False]

ParametricPlot[ cc[u, 1], {u, 0, 2 Pi}, Axes -> False, PlotStyle -> Thick,
ColorFunction -> Function[{x, y, u}, Hue[u/(2 Pi)]],
ColorFunctionScaling -> False]

and to plot a hyperbola we'd use Exclusions
(to get rid of zero from the denominator) e.g.
ParametricPlot[ cc[u, 3/2], {u, 0, 2 Pi}, Axes -> False, PlotStyle -> Thick,
ColorFunction -> Function[{x, y, u}, Hue[u/(2 Pi)]],
ColorFunctionScaling -> False, Exclusions -> {Cos[u] == 2/3, Cos[u] == -2/3}]

ParametricPlot[]
. $\endgroup$ – J. M.'s ennui♦ Jun 12 '13 at 16:38