In the plane, the usual notation is $$\big(x(t),y(t)\big)$$ for the function and $t$ for the parameter.
A function $f(x)$ is a parametric function with $x(t)=t$ and $y(t) = f(t)$.
The parametric equation for the circle is $$x(t) = \cos t,\quad y(t) = \sin t,\quad t\in\R.$$ This is seen immediately as $x^2(t)+y^2(t) = 1$ for all $t$.