Show two plots together: a two dimensional curve tangent to the maxima of a three dimensional plot

peakValues01 = 
  Pick[yequalto01, PeakDetect[yequalto01[[;; , 2]]], 1][[;; -2]];
peakValues126 = 
  Pick[yequalto126, PeakDetect[yequalto01[[;; , 2]]], 1][[;; -2]];

p01 = Join[{ConstantArray[0.01, Length[peakValues01]]}, 
    peakValues01\[Transpose]]\[Transpose];
p126 = Join[{ConstantArray[1.26, Length[peakValues126]]}, 
    peakValues126\[Transpose]]\[Transpose];


Show[
 ListPlot3D[data3D],
 ListPointPlot3D[p01],
 Graphics3D@Line@p01,
 ListPointPlot3D[p126],
 Graphics3D@Line@p126
 ]

Of course, you can customise the lines/dots as you want with the standard options. If you want smooth lines between the points, I would try with interpolating first and then a ParametricPlot:

p01f[x_] = Interpolation[peakValues01][x];
p126f[x_] = Interpolation[peakValues126][x];

htl = Join[{#*\[Pi], #*\[Pi], {0.014, 0}} & /@ 
    Range[0, 8, 2], {#*\[Pi], "", {0.01, 0}} & /@ Range[0, 8, 1]];

Show[
 ListPlot3D[data3D, ColorFunction -> "TemperatureMap", Mesh -> None
  , Ticks -> {Automatic, htl, Automatic}, BoxStyle -> Dashed, 
  AxesLabel -> {"y", "x"}],
 ListPointPlot3D[p01, PlotStyle -> {Purple}],
 ListPointPlot3D[p126, PlotStyle -> {Blue}],
 ParametricPlot3D[{0.01, x, p01f[x]}, {x, 0., 25.1}, {y, 0, 1.26}, 
  PlotStyle -> {Purple, Thickness[0.02]}],
 ParametricPlot3D[{1.26, x, p126f[x]}, {x, 0., 25.1}, {y, 0, 1.26}, 
  PlotStyle -> {Blue, Thickness[0.02]}],
 ViewPoint -> {4, 1, 1}
 ]

{y01, y126} = Table[Select[data3D, #[[1]] == i &], {i, {.01, 1.26}}];

{peaks01, peaks126} = Pick[#, PeakDetect[#[[;; , 3]]], 1] & /@ {y01, y126};

Show[ListPlot3D[data3D], 
  Graphics3D[{PointSize[Large], Thick, Red, Line @ peaks01, Point @ peaks01,
   Green, Line @ peaks126, Point@ peaks126}]]