Parameter

The interval of parameter values belongs to the curve rather than sitting outside it. With $x = t$ and $y = t^{2}$ over all real $t$ you get the whole parabola $y = x^{2}$, while $x = t^{2}$ and $y = t^{4}$ satisfy the same relation but produce only the half with $x \ge 0$, and cover it twice as $t$ sweeps the reals.

In motion problems the parameter is time, which makes derivatives taken with respect to it physical: $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are velocity components, and dividing $\frac{dy}{dt}$ by $\frac{dx}{dt}$ gives $\frac{dy}{dx}$, a slope with the time scaled out. A polar curve is the same setup with the angle $\theta$ as the parameter.

Eliminating the parameter trades the extra variable for one relation between $x$ and $y$. You keep the shape and lose the direction, the timing, and often part of the curve, so the resulting equation is a sketching aid rather than an equal description.