I think many people are familiar with the Folium of Descartes:
$$ x^{3}+y^{3}-3axy=0\,.$$
Choosing $\; p =\frac yx\; $ produces a rational parametric equation:
$$ x = \frac{3at}{1+t^3}; \quad y = \frac{3at^2}{1+t^3}$$
that after plotting results in a continuous curve.
Whereas WA produces this obviously incorrect plot (here $ a = 1 $, but that doesn't really matter).
The reasoning behind this is beyond my comprehension, but I'm afraid it can be a sign of a more major issue with parametric plots.
There are fewer issues with the plot of the polar version of the same equation, yet the asymptote $\; y = -x -a \;$ is still present as a part of the curve for some reason.
What is happening here?
This is because WolframAlpha is plotting for $-5.784 \leq t \leq 2 \pi$. Apparently it deems this to be the most useful plot range, for reasons unknown to me. The asymptote appears as it attempts to plot over any range containing $0$, where it decides it needs to zip over from one side of the graph to the other. I agree this is suboptimal.
I don't know how to get WA to plot from $-\infty$ to $\infty$, I'm afraid, but you can get a better result by telling it to plot between larger values.