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]
