Question: I'm studying statistics, and in a lot of textbooks, the regression formulas always refers to the functions themselves as f(x) parameterized by a,b,c or something. And they are often written $f(x; a,b,c)$, why can't we just write $f(x, a, b, c)$?

I would like to know perhaps the history of how this came about, and why there is this seemingly dual method of expressing the same idea. Is there some esoteric mathematical notation that a noob like me hasn't encountered yet, or did mathematicians of old just like doing things in different ways?

An example: A function that maps age of tree to height:

$$f(age; growth\\_rate) = age \* growth\\_rate$$ vs $$f(age, growth\\_rate) = age \* growth\\_rate$$

Answer: 
While the answers already given are correct, I suspect that they may be bringing in terminology and notations you are unfamiliar with. If so, let me try to recast the answer more simply.

You are correct that $f(x; a, b, c)$ is the exact same thing as $f(x, a, b, c)$. But the difference is in how we are using $x$ vs how we are using $a, b, c$. With $f(x, a, b, c)$, all four values are equally important, equally of interest. With the former, we are considering $f$ as a function of $x$ only. $a, b, c$ are simply values that we use in defining that function. By making these values variables too, the mathematics we do applies to all the different functions of $x$ that are defined by the various possible values of $a, b, c$, instead of having to work each particular example we need separately.

For example, when talking of linear functions, we will write things such as $$y = mx + b$$ You could consider this as defining $y$ as a function of three variables, $x, m, b$, but what is important is the dependence of $y$ on $x$. The slope $m$ and intercept $b$ are just values expressing which line is represented by the relationship between $y$ and $x$. So if want to find the root for $y = 0$, that is $x = \frac{-b}{m}$, not $b = -mx$ or $m = \frac{-b}x$.

That is what the notation $f(x; a, b, c)$ means. It is just telling you that $x$ is the one whose variance is important to us. The others we are going to consider fixed (although not currently known) values.
The area where I am used to seeing parameterizations is when considering curves and similar geometric objects. Consider a curve drawn in $\mathbb{R}^3$, this is just a set of points and initially is not a function of any kind. But we can in fact make it into the image of a function by defining a mapping $f:[0,1] \rightarrow \mathbb{R}^3$ which 'traces' the curve as the input variable goes from 0 to 1. So in this context a parameterization means taking an object that is just a subset of a given space and defining a mapping whose image is the object we are interested in. The inportant thing here is that the choice of this mapping is arbirtrary, in that there are an infinite number of different functions we could define whose image would be our object of study. Hence calling it a parameterization really is indicating that we have chosen a suitable functional description, but indicates that it is not unique.

This seems the same as for the function example you give, as there are any number of ways of taking the heights of trees and defining functions which map to these correctly for each tree. So your semicolon notation appears to simply be saying that this function has been chosen to use growth rate along with the fixed age input in order to define height, but is indicating that there were other choices you could have made instead.