I am adding four usually requested options for .GIF, so that everything is in one post: the number of repetitions, the pause between each repetition, the animation time, and the direction of the animation.

frame[t_] := Show[
   ParametricPlot[{Sin[3 x], Sin[4 x]}, {x, 0, t}, Axes -> None, 
    PlotStyle -> Thick, PlotRange -> {{-1.05, 1.05}, {-1.05, 1.05}}, 
    ImageSize -> Large],
   Graphics[{PointSize[Large], Red, , Point[{Sin[3 t], Sin[4 t]}]}]
   ];
framesForward = Table[frame[t], {t, 10^-6, 2 \[Pi] + 10^-6, \[Pi]/50}];
framesBackward = Reverse[framesForward];
framesForwardBackward = Join[framesForward, framesBackward];

Export["Lissajous forward.gif", framesForward, 
  AnimationRepetitions -> \[Infinity], 
  "DisplayDurations" -> 
   ReplacePart[
    Table[5/Length[framesForward], Length[framesForward]], -1 -> 2.0]];

Export["Lissajous forwardbackward.gif", framesForwardBackward, 
  AnimationRepetitions -> \[Infinity], 
  "DisplayDurations" -> 
   ReplacePart[
    Table[10/Length[framesForwardBackward], 
     Length[framesForwardBackward]], {Length[framesForward] -> 
      2.0, -1 -> 2.0}]];

References

https://mathematica.stackexchange.com/questions/115851/add-delay-to-the-final-frame-of-a-gif?rq=1 https://mathematica.stackexchange.com/questions/145536/create-gif-looping-animation Mathematica's documentation for .GIF: https://reference.wolfram.com/language/ref/format/GIF.html