Assuming that your 3D data is in an array named dat and your common parameter for all three is in an array called param then you can write:
interp = Table[Interpolation[Transpose[{param, dat[[All, i]]}]], {i, 3}]
That will give you a table of interpolation functions for each of the three values in your dat array. Then you can write:
ParametricPlot[{interp[[1]][t], interp[[2]][t]}, {t, Min[param], Max[param]},
PlotStyle > Thick,
ColorFunction > Function[{x, y, t}, ColorData["Rainbow"][interp[[3]][t]]],
ColorFunctionScaling > False]
To plot your colored curve.
