If 5 points are necessary to determine a conic, why are only 3 necessary to determine a parabola?

Solution 1:

Illustrating a comment to OP's own answer.

Shown is an entire family of parabolas through three points, indicating that three points alone do not determine a parabola.

As mentioned in my comment, a conic in the coordinate plane has five characteristics: eccentricity, scale, orientation, $x$-location, $y$-location. Our interest in parabolas specifies a characteristic (eccentricity $=1$); the points account for three more characteristics; and the animation cycles through the aspects of the fifth (here, orientation).

Solution 2:

As you can see from the image below, if you are defining a conic, it requires five points to be known. The conics below all share three points (black), but need two more to define what they are.

If we trying to simply define a parabola, it only takes three points because we know it is a parabola, and it can't possibly be a ellipse or hyperbola -- because we decided we already that we are making a parabola!