A parametric representation of a curve $\gamma\subset{\mathbb R}^2$ is an at least continuous function ${\bf f}:\quad t\mapsto {\bf z}(t)=\bigl(x(t),y(t)\bigr)\qquad(a\leq t\leq b)\ .$ A priori the independent variable $t$ is not required to have any geometrical interpretation and can be thought of as "time".
It is a fact of life that any curve $\gamma$, which is a static or "drawn" geometrical object, has many equivalent parametric representations, among them some where the independent variable has a geometric meaning connected with the curve; e.g. it could be the $x$-coordinate of the running point, the arc length along the curve, measured from a starting point, etc. One such "geometrical" variable could be the changing tangential angle $\theta={\rm arg}(\dot x,\dot y)$, but it is not suitable for all curves. It is definitely necessary that the above function ${\bf f}$ is well defined, and this means that to any $\theta\in{\mathbb R}$ (or $\in{\mathbb R}/(2\pi)$) should correspond at most one point of the curve having this tangential angle $\theta$. Such is the case for strictly convex curves, but not for curves containing pieces of lines or for a sinusoid $x\mapsto(x,\sin x)$ $\ (-\infty.