The general equation of a conic section is ax^2 + bxy + cy^2 + dx + ey + f = 0
The equation given above has 5 coefficients. To determine all of them at least 5 equations are required.
A parabola can open along the y-axis and this has a general equation y - h = a*(x - k)^2 or it can open along the x-axis and this has a general equation x - h = a*(y - k)^2. If it is given in which direction the parabola opens it would be possible to identify a unique parabola passing through 3 points. Else, the co-ordinates of at least 5 points should be given.
A general parabola is not uniquely defined by 3 points.