Change plot style as function of parameter in ParametricPlot

Clear["Global`*"]

f[t_ /; 0 <= t <= 1/2] := ReIm[Exp[2 \[Pi] I t]]
f[t_ /; 1/2 < t <= 1] := ReIm[Exp[2 \[Pi] I (1 - t)]]

You could use ColorFunction

EDIT:

Manipulate[
 ParametricPlot[f[t], {t, 0, v},
  PlotRange -> {{-1, 1}, {-1, 1}},
  ColorFunction -> (If[#3 < 1/2, Blue, Red] &),
  ColorFunctionScaling -> False],
 {{v, 0.01}, 0.01, 1, 0.01, Appearance -> "Labeled"}]

The plot of a single function without singularities will result in a single Line object, which can only a single thickness. To get two thicknesses, you would need to draw two lines. This may or may not be acceptable, since it doesn't look like a single function.

Either

Manipulate[
 ParametricPlot[{
   Style[ConditionalExpression[f[t], t <= 1/2], AbsoluteThickness[1], Blue],
   Style[ConditionalExpression[f[t], t >= 1/2], AbsoluteThickness[3], Red]},
  {t, 0, v},
  PlotRange -> {{-1, 1}, {-1, 1}},
  PlotRangePadding -> Scaled[0.01]],
 {{v, 0.01}, 0.01, 1, 0.01, Appearance -> "Labeled"}]

or

Manipulate[
 ParametricPlot[{
   ConditionalExpression[f[t], t <= 1/2],
   ConditionalExpression[f[t], t >= 1/2]},
  {t, 0, v},
  PlotRange -> {{-1, 1}, {-1, 1}},
  PlotStyle -> {
    Directive[AbsoluteThickness[1], Blue],
    Directive[AbsoluteThickness[3], Red]},
  PlotRangePadding -> Scaled[0.01]],
 {{v, 0.01}, 0.01, 1, 0.01, Appearance -> "Labeled"}]