Can every parabola be written in the form of a quadratic $y=ax^2+bx+c$ or $x=dy^2+ey+f$? I understand that the graph of any equation of the form $y=ax^2+bx+c$ is a parabola (please correct me if I am mistaken).
My question is about the converse: Can every parabola be written in the form of an equation $y=ax^2+bx+c$ or $x=dy^2+ey+f$? 
 A: A parabola is just the locus of points (a collection) that is equidistant from a point and a straight line. Let $(x,y)$ be the set of points, then for let's say you have a fixed point $(p,q)$ and a line $ax+by+c=0$, 
The distance between $(x,y)$ and the straight line is given by:
$$\frac{|ax+by+c|}{\sqrt{a^2+b^2}}$$
We want to set that to be equal to the distance between $(p,q)$ and $(x,y)$
$$ \frac{|ax+by+c|}{\sqrt{a^2+b^2}}=\sqrt{(x-p)^2+(y-q)^2}$$
$$\frac{a^2x^2+b^2y^2+2abxy+2acx+2bcy}{a^2+b^2}=x^2-2px+p^2+y^2-2qx+q^2$$

$$b^2x^2+a^2y^2-(2p(a^2+b^2)+2ac)x-(2q(a^2+b^2)+2bc)y-2abxy+(p^2+q^2)(a^2+b^2)=0$$
So the formula doesn't look pleasant, but we know from that, any parabola will be in the form of $ax^2+by^2+cx+dy+exy+f=0$
A: The parabolas whose axes of symmetry is parallel to x- or y- axis belong to the category you mention. They are characterized by an absence of $xy$ term which when present, makes its axis of symmetry inclined to the coordinate axes.
From the general equation of conics using an invariance condition $I_2=0$ for unity eccentricity  of parabola we have
$$ a x^2 + 2  h x y + b y^2 + 2 g x+ 2 f y + 1 = 0,\, h^2 - a b =0 ;$$
when solved for $y$ as a quadratic and constants are re-lumped, it assumes a form for the inclined parabola:
$$  \boxed {y = (A x + B) \pm \sqrt{ C x + D } } $$
The two branches are separated by vertical/horizontal tangents when $ C x + D =0 $

A: To sum up the comments: No.
But you're not completely wrong. Parabolas are "just" geometric shapes, they don't require a coordinate system, but if you add a coordinate system with axes parallel to the directrix and the axis if symmetry (see Wikipedia in parabolas) if can be described by one of the two forms you have listed.
A: If it is any parabola in the plane you would need $$ax^2+by^2+cxy+dy+ex+f = 0$$, this becomes evident if you try and for example rotate your parabola:
$$x_0 = \cos(\alpha)x + \sin(\alpha)y\\y_0 = -\sin(\alpha)x + \cos(\alpha)y$$
You will be getting the "mixed terms" containing $xy$ when converting $x^2$ and $y^2$ into $x_0$ and $y_0$. Some special $\alpha$ maybe pose no problem but for most of them you will be getting the cross-terms.
