Claude,
You put the inset in the wrong place. Adjust the x,y coordinates a bit. I also made the main plot bigger to get more room for the plot and adjusted the vertical to align the axis:
ParametricPlot[{Sin[x], Sin[2 x]}, {x, 0, 4 \[Pi]}, Frame -> True,
ImageSize -> Large, Prolog -> Inset[p1, {.56, -.02}]]
gives you
Regards,
Neil
