Polynomial parametrization and degree nominator of parametrization of a rational curve

Question

Let $\mathcal{C}\subset\mathbb{R}^2$ given by $f(x,y)=0$ be a rational curve. Then, there exists a rational parametrization $\phi(t)\,:\mathbb{R}\rightarrow \mathbb{R}²$. Under which circumstances is this parametrization polynomial? For example, if the equation defining the curve is of the from $h(x)-y=0$ then this is the case. But are there other instances? Moreover, can one get estimates on the degree of the denominator which is necessary?