How many parabolas can pass through two given points? I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.
My question is, how many? 
 A: Infinitely many.
Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.
A: Any size of parabola, with infinitely many orientations for each parabola!
Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.
Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)
Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,\infty)$ and an angle in the range $(0,π)\cup(-π, 0)$.
Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.
A: General equation of a conic section:
$\small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$; 
where $A,B,C,D,E,F$ are constants.
Necessary condition for a parabola :
$B^2-4AC =0$, which leaves $5$ constants to be determined.
Cf.  Blue's link in his comment above.
