Why do parabolas' arms eventually become parallel? Here is what this site states

All parabolas are the same shape, no matter how big they are. Although they are infinite, meaning that the arms will never close up, the arms will eventually become parallel.

Now, I have an argument against it. Let $f(x) = ax^2 + bx + c$ be a quadratic polynomial with $a , b$ and $c$ being real numbers and $a \ne 0$. So, its graph will give us a parabola ($\because$ graph of a quadratic polynomial is a parabola). 
Now,
$$\dfrac{d (ax^2 + bx + c)}{dx}$$$$ = 2ax + b$$
i.e. the slope of a quadratic polynomial is given by $g(x) = 2ax + b$. Now, differentiating the equation for slope of the quadratic
$$\dfrac {d (2ax + b)}{d x}$$
$$ = 2a$$
So, if $a \gt 0$ then the slope of $g(x)$ will be increasing. This means that the slope of $f(x)$ will also be increasing. Similarly, if $a \lt 0$ then the slope of $f(x)$ will be decreasing. 
This means that for all $x$ the slope of $f(x)$ will be different. So, this contradicts the fact (according to me) that the arms of a parabola will eventually be parallel. Where am I going wrong?

There is no contradiction. As you said, the slope keeps increasing, this means that as you go towards $x\rightarrow\pm\infty$, the slope also goes to $\pm\infty$, i.e the graph of the function gets more and more "vertical".
