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I am writing some code in which I combine a ParametricPlot[] with a RegionPlot[]. Mathematica is very good at figuring out appropriate bounds for the ParametricPlot[], and I would like to reuse the same bounds for creating the RegionPlot[]. How can I tell Mathematica to reuse the bounds?

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If you mean PlotRange, then the function completePlotRange in this answer to How to get the real PlotRange using AbsoluteOptions? might be of interest. – Michael E2 Nov 23 '13 at 22:22
@MichaelE2 Doesn't seem necessary in this particular case – Dr. belisarius Nov 23 '13 at 22:34
@belisarius I noticed, but I still thought it was a good reference for others, as well as a possible duplicate. – Michael E2 Nov 24 '13 at 0:15
up vote 2 down vote accepted
Show[RegionPlot[x + y < 1/2, Evaluate[Sequence @@ MapThread[Join, {{{x}, {y}}, (PlotRange /. 
                                                            AbsoluteOptions[#, PlotRange])}]]], 
     #] &@ ParametricPlot[{Sin[u], Sin[2 u]}, {u, 0, 2 Pi}]

enter image description here

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The construction (PlotRange /. AbsoluteOptions[#, PlotRange]) can in fact be substituted, I think, by the much simpler PlotRange[#]. As far as I can tell this is not documented in the PlotRange documentation, but it does seem to work. – Emilio Pisanty Aug 20 '14 at 15:30
@episanty Efectivamente. I learnt about that in a recent answer here. Thanks! – Dr. belisarius Aug 22 '14 at 16:10

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