DynamicModule[{tr = 1},
Column[{
Show[
ParametricPlot[
{Cos[t], Sin[2 t ]}, {t, 0, 2 Pi},
GridLines ->
Dynamic[List /@ CurrentValue[{"MousePosition", "Graphics"}]]
, PlotStyle -> Directive[Thin, Gray]
],
Graphics@{Dynamic[First@ParametricPlot[
{Cos[t], Sin[2 t ]}, {t, 0.0001, 2 Pi tr},
PlotStyle -> Thick
] /. Line -> Arrow]
}
, ImageSize -> Large],
Slider@Dynamic@tr
}]
]