I take V as the origin of local coordinates . F must then lie on the line through x2. P is a free point in space.
One point controls one property:
Translation (Vertex V),
Rotation (A),
Scale/Shape* (F or P).
If F is moved, P is not uniquely determined. I choose to hold k constant, and vary h. In words, F changes the width of the parabola.
Now, F and P have the same job. This is a good time to introduce a control structure:

The control diagram relates GGB objects V, A, F, P, p, h, k.
Now, it is possible to determine the parabola locus using only two points. Say, V and F. Suppose I had done this. There are two problems:

Unwieldy. F change both the shape and rotation of the curve. This does not help me construct measured figures, although sometimes it is pretty.

Not enough information to orient the curve. For example:Does the parabola open up or down? What is the direction of increasing t?
Direction in each dimension is a free choice. There is no answer. We must give the direction of increasing t (here, x1).
The UP direction (x2).

*On a parabola, shape and scale may be indistinguishable. I leave it this way for now. I will separate the two when it describes the problem at hand.