Question

for most pairs of parametric equations the parameter t represents the time (in a unit such as seconds or days) after a particular event occurs. Why is time such an appropriate parameter?

Answer 1

Most equations we deal with in mathematics and physics can be though of as position functions. We use time to describe the motion of a particle and describe this motion with a function.

One good reason to choose time is that it's no (usually) possible for a particle to occupy the 2 different points in space at the same time.