ClearAll[x, t, s];
Manipulate[
  curves = {x[t, #], y[t]} & /@ s;

  ParametricPlot[curves, {t, 0, 1}, ImageSize -> 300,
        PlotRange -> {{-2, 2}, Automatic}, PlotLegends -> "Expressions"],

  {{s, Range[0, 1, .2], "s"}, Range[0, 1, .2], ControlType -> TogglerBar},

  Initialization :> (
   x[t_, s_] := (Print[s]; t - Exp[t] + 2 s);
   y[t_] := Exp[t]
   )
 ]