What is a the intuition behind a parametric equation? I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I used before. Of course, I can't understand the plane and the parametric equations for it. Any good example or reference for having a good intuition about them. I'm a first year student in economics, if possible, an example in economics will be of great help.
I recently saw a video on khan academy about parametric equations where both x and y depend on time, time being the parameter. Why not just make a 3d graph with 3 variables: x, y and t instead of a parametric equation in 2d?

A set of parametric equations is called by that name because they have a parameter.  Such equations are often used when a graph might not have certain "typical" and/or "nice" properties like being one-to-one or onto, e.g., a circle.
For a circle, a set of parametric equations might look like this:
$$y=a\sin t$$
$$x=a\cos t$$
Then the circle of radius $a$ (centered at the origin) is generated by varying the value of $t$ and applying that value within each of the equations.
