How to color the same contour line with different colors according y value

If it is acceptable to order the lines by the y value of the first point in each contour we can use a modification of Pickett's method:

plot = Show[{ContourPlot[Sin[y - x^2] == 0, {x, -1, 0}, {y, -10, 10}], 
   ContourPlot[Sin[-y - x^2] == 0, {x, 0, 1}, {y, -10, 10}]}, PlotRange -> All];

cols = {Red, Black, Blue, Orange, Green, Pink, Brown};

Normal[plot] /. {a___, l : Longest[Line[_] ..], b___} :>
  {a, Riffle[cols, {l} ~SortBy~ Extract[{1, 1, 2}] ], b}

Or using xslittlegrass's restylePlot2 as the base method:

restyleWithSort[p_, op : OptionsPattern[ListLinePlot]] :=
  ListLinePlot[Cases[Normal@p, Line[x__] :> x, ∞] ~SortBy~ Extract[{1, 2}], 
   op, Options[p]]

Show[restyleWithSort /@ {
   ContourPlot[Sin[y - x^2] == 0, {x, -1, 0}, {y, -10, 10}],
   ContourPlot[Sin[-y - x^2] == 0, {x, 0, 1}, {y, -10, 10}]
   },
 PlotRange -> All
]

You can manipulate the expression manually:

plot = ContourPlot[
   Sin[y - x^2] == 0, {x, -1, 1}, {y, -10, 10}
   ];

plot /. {a___, l : Longest[Line[_] ..], b___} :>
  {a, Riffle[Array[ColorData[97], Length@{l}], {l}], b}

In order to find the pattern it helps to look at plot // FullForm. ColorData[97] is the list of default colors for Mathematica 10. Here is another answer I wrote two days ago, which also shows how this technique can be used.

func[cplot_, cf_, s_] := With[{cp = cplot},
  pt = cp[[1, 1]];
  lines = Cases[cp, Line[x__] :> x, -1];
  max = Max[#[[All, 2]]] & /@ (Part[pt, #] & /@ lines);
  Show[Graphics[
    GraphicsComplex[pt, 
     MapThread[{cf[Abs[Rescale[#1, {Min[max], Max[max]}] - s]], Thick,
         Line[#2]} &, {max, lines}]], Frame -> True, 
    AspectRatio -> Full]]
  ]

For this example:

p1 = ContourPlot[Sin[y - x^2] == 0, {x, -1, 0}, {y, -10, 10}];
p2 = ContourPlot[Sin[-y - x^2] == 0, {x, 0, 1}, {y, -10, 10}];
Manipulate[
 Show @@ (func[#, color, s] & /@ {p1, p2}), {{color, 
   Hue}, {Hue -> "Hue", ColorData["Rainbow"] -> "Rainbow", 
   ColorData["Pastel"] -> "Pastel"}}, {{s, 0, "invert"}, {0, 1}}]