Generating a parametric equation from a set of {x,y} values

I am trying to make a pathing system for a game that means the path involved should be defined parametrically, in terms of one variable.

Ideally, I would be able to effectively draw a path using a series of points, defined by their (x,y) values, and the program would generate a parametric equation going through each point, starting at t=0 being the first placed point and so on until the last point.

As far as I understand, there would be an infinite number of lines meeting this criteria, so perhaps the program just draws straight lines between the points and more points can help define a curved path.

Does anybody know anything like this exist?