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?
By "eventually" they mean "in the limit". As $a$ goes to positive infinity, the slope at $a$ goes to positive infinity, which means the arm is becoming more and more vertical. As $a$ goes to negative infinity, so does the slope at $a$ - but a slope of "negative infinity" also means that the line is vertical, so the left arm is also becoming closer and closer to vertical. If both arms are getting closer to closer to vertical, then they must be getting closer and closer to being parallel to one another.

Of course, your argument shows that they will never actually be parallel. Just very close.
Obviously, the tangents become vertical. Verticals are parallel.