Note that unlike the circle, which is smooth, the folium has a singular point at the origin where the curve crosses itself. For curves or surfaces defined by a single polynomial equation, singular points occur when the Taylor expansion of the polynomial about that point contains no contant or linear terms. Equivalently, a point is singular if the polynomial and all of its first-order partial derivatives vanish at the point. In our case, it is clear that (0,0) is a singular point.
Take an arbitrary line through the singular point, namely y = mx. In general, we would expect this line to meet our cubic curve in three distinct points, but because the origin is singular, we will get a double root there and the remaining root will necessarily be rational:
x^2[(1+m^3)x - 3m] = 0
so x = 0 (our double root) or x = 3m/(1+m^3) and y = 3m^2/(1+m^3)
giving a rational parameterization of our curve.
Exercise: Find integer solutions to the equation
by first converting it to an equivalent equation having rational solutions, then using the techniques of this section [the singular point is not at the origin in this case].