Arrow head halfway along parametric plot

The real problem your are having is the way GraphicsComplex uses VertexColors. In short, your parametric plot with colors produces something like this

Graphics[GraphicsComplex[{{0, 0}, {1, 1}, {2, 1}},
  {Line[{1, 2, 3}, VertexColors -> Automatic]}, 
  VertexColors -> {Yellow, Blue, Green}
  ]]

Note that at the end of Line it says VertexColors -> Automatic which states that it uses the vertex color definition that is given at the end of GraphicsComplex.

Just replacing Line with Arrow will therefore not work and you will get an error message because Arrow does not take a VertexColors option.

In fact, Arrow cannot be used with VertexColors at all. Fortunately, Arrow cannot take only points as arguments, but Lines primitives itself. Therefore, one solution to your problem is

p0 = ParametricPlot[{t, t^2}, {t, 0, 1}, PlotStyle -> Thick, 
  PlotRange -> {{-0.8, 0.8}, {-0.8, 0.8}}, 
  ColorFunction -> Function[{x, y, t}, Hue[0.7*t]], Frame -> True, 
  FrameLabel -> {{"y/\[Lambda]", None}, {"x/\[Lambda]", 
     HoldForm["f" = Fi]}}, 
  GridLines -> {{0.4, 0, -0.4}, {-0.4, 0, 0.4}}, 
  FrameTicksStyle -> Thick, 
  FrameTicks -> {{{-0.8, -0.4, 0, 0.4, 0.8}, 
     None}, {{-0.8, -0.4, 0, 0.4, 0.8}, None}}, 
  GridLinesStyle -> Opacity[0.5], Axes -> False, ImageSize -> 300, 
  PlotRangePadding -> None, BaseStyle -> {FontSize -> 18}, 
  AspectRatio -> Automatic]

p0 /. l_Line :> {Arrowheads[{0, 0.1, 0}], Arrow[l]}

In the absence of definitions for x, y, and pt, set

 x = t^2; y = t^3;

and replace p[[1]] by 1 for specificity. Then an arrow is obtained by replacing the last line of the code by

(p0 // Normal) /. Line[x_] :> {Arrowheads[{0, 0.1, 0}], Arrow[x]}

Normal is needed to convert GraphicsComplex to Line.