The function Show is tricky, because it will use the PlotRange of the first plot and ignore the other ranges, unless you give it a value at the end. What plot range do you wish for the combined plot?

Show[
 ParametricPlot[{r^2 + 1, 1/r}, {r, 1, 5}],
 ParametricPlot[{r^2 + 2, 2/r}, {r, 1, 5}],
 PlotRange -> All, AspectRatio -> 1/2]

or, more simply,

ParametricPlot[{{r^2 + 1, 1/r}, {r^2 + 2, 2/r}}, {r, 1, 5},
 PlotRange -> All, AspectRatio -> 1/2]

I don't know of a straightforward way. You either change the ranges of r, or use RegionFunction:

Show[ParametricPlot[{r^2 + 1, 1/r}, {r, 1, 5},
  PlotRange -> {{0, 10}, {0.2, 0.8}},
  RegionFunction -> Function[{x, y}, 0.2 < y < .8]],
 ParametricPlot[{r^2 + 2, 2/r}, {r, 1, 5},
  RegionFunction -> Function[{x, y}, 1 < y < 2]],
 AspectRatio -> 1/2, PlotRange -> {{0, 10}, {0.2, 2}}]