If one wants to parametrize the curve of intersection in $\mathbb{R}^3$ of the surfaces $y=x^2$ and $z=x^3$, would it be correct to parametrize this curve as $\bf{g}$$(t)=(t, t^2,t^3)$?
The reason I'm asking this is because sometimes there are subtle errors in what we think should be true.
Yes, your parametrization is correct. The first surface could be parametrized by ${\bf x}(u,v) = (u, u^2, v)$. If we're in the intersection, we must have $v = u^3$. So the curve only has one parameter now, which I'll call $t$, resulting in $(t,t^2,t^3)$.