As Sander points out, there are many varieties of ParametricPlot, 2D or 3D, one variable or two, but here is just an idea of how you could build a ColorFunction to to suit your needs:

{ParametricPlot3D[{Sin[u], Cos[u], u/10}, {u, 0, 20}, 
  ColorFunction -> Function[{x, y, z, u}, Hue[u]]],
 ParametricPlot3D[{4 + (3 + Cos[v]) Sin[u], 4 + (3 + Cos[v]) Cos[u], 
   4 + Sin[v]}, {u, 0, 2 \[Pi]}, {v, -\[Pi], \[Pi]}, 
  ColorFunction -> Function[{x, y, z, u, v}, Hue[u + v]]],
 ParametricPlot[{Sin[u], Sin[2 u]}, {u, 0, 2 Pi}, 
  ColorFunction -> Function[{x, y, u}, Hue[u]]]}