The slope from (0,0) to (2,3) is 3/2. The slope of the line tangent to the parabola at x=13/10 (the vertex) is 0. These are clearly not equal.
Given 3 points on a parabola, the slope at the vertex will not always equal the slope of the secant line between two of those three points. Hopefully you can see from the graph of a parabola what the slope is at the vertex (hint: it's always going to be zero). It turns out that the only time a secant line is going to have the same slope as the line tangent to the vertex is when the points are equidistant from the vertex horizontally.