An arrow is often put on the curve to indicate the direction of increasing time or orientation of the curve. The parameter does not always represent time. Here, the parameter t represents the polar angle of the position on a circle of radius 3 centered at the origin and oriented counterclockwise. It is often possible to re-write the parametric equations without the parameter. When you do eliminate the parameter, always check that you have not introduced extraneous portions of the curve.
Every curve has infinitely many parametrizations, amounting to different scales for the parameter.
This is particularly useful when neither x nor y is a function of the other.