Calculating Parabolic Curves in 3D Space
Jelly, 13 11 bytes
_÷/µḢ²=A;$P

The input format is a bit strange; it takes P1 as the second argument, and (P3,P2) as the first argument, where each point is expressed in the form ((x+yj),z).
We first normalize the points so that the vertex is on the origin, and then make sure that P1 and P2 are on the same line, and also that the ratio of their z distances is equal to the ratio of their x distances.
_÷/             Helper link. Inputs: (P3,P2) on left, P1 on right.
_                  Subtract right from left. (P3-P1,P2-P1). 
 ÷/                Fold division. (P3-P1)/(P2-P1). P2-P1 is nonzero in both x+yi and z.
   µ               Push the chain onto the stack, and start a new monadic link.
    Ḣ²=A;$P     Hook; main link. Calculates whether w² = a and a is nonnegative.
                Input: The result of the helper link. Call it [w,a].
    Ḣ²              Pop w and square it. Now the list is [a].
       A,$          Concatenate a to its absolute value.
      =             Fork; is w equal to [a,abs(a)]? Returns 2 element list of 0 or 1
          P         Product; that is, true if w was equal to both a and abs(a).

Try it here.