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parametric plot and filling

Dear group:

DoesParametricPlot admit the Filling option? Seems it not? How then to perform it?
Lokk at the following simple example:

ParametricPlot[{{2 t,-t^2},{6+ 2 t,-t^2}}, {t,-2,2},Axes->{True,False}, Ticks->None, PlotStyle->{{Thick,Cyan},{Thick,Green}}]

The if I introduce the Filling command inside PlotStyle nothing happens, and outside it produces an error. Any ideas?

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