What is a variable as opposed to a parameter?
I have never seen or heard of a clean definition.
As of my opinion, a parameter is a fixed value, whereas a variable is, deep thought, the result of an undefined function, like it is for "stochastic" variables, minus that "stochastic" prefix.
Mathematical expressions involving many letters and other symbols are often used to define functions, such as $$ax^2+bx+c.$$
Writing $n$ for the number of distinct letters involved (four in the example above), the domain of the function one has in mind is usually not as big as $\mathbb R^n$, but rather $\mathbb R^m$, where $m$ is the number of a chosen subset of letters which are called the variables, while all others are called parameters.
Of course, in order to define a concrete function, the parameters must be assigned fixed values, with different choices leading to different functions.
Speaking of the example above, a popular choice is to call $x$ the variable and $a$, $b$ and $c$ the parameters, but nothing prevents us from making other choices.
A parameter is a constant that can be varied if desired. Variables depend on parameters that can be changed.
$$ (x,y)= (2ft,ft^2) $$
is a parabolic curve equation where any point can be associated with a parameter $t$. When $t$ is eliminated we have a single equation $ 4 f y= x^2. $
Several parameters can be used to distinguish parabolas of different focal lengths $f$.
