@Mariah Yes, a function f(x) is single-valued so the graph y = f(x) has certain restrictions that exclude, for instance, horizontal parabolas. But @PhiNotPi is asking about all parabolas that pass through two points. This need not be restricted to the form y = f(x).
To give a different example, the graph of x^2 + y^2 – 1 = 0 is a circle. It is not a single-valued function y = f(x). But you can consider G(x,y) to be a function of two variables where G = x^2 + y^2 – 1 whose graph G=0 is a unit circle centered at the origin.
I don’t remember the general form of a parabola and if I had time I could work out a transformation of axes allowing a vertical parabola (such as you described earlier) to be transformed to a tilted parabola whose axis is along any arbitrary line in the plane, in which case G(x,y) will contain xy terms—this is true of any conic section..
Speaking of planes, however, I am on the eve of international travel & might not have time to work out a more complete answer! (Hate when that happens.) I know that’s a cheap cop-out, though it happens to be true. Maybe somebody else could work it out.