Answer
$x^2=8y$, $(-4,2),(4,2)$
See graph.
Work Step by Step
1. Given the focus at $(0,2)$, vertex at $(0,0)$, we can write the parabola equation in a standard form $4py=x^2$. With $p=2$, we get $x^2=8y$
2. Let $y=2$, we have $x^2=16$ and $x=\pm4$ and the two points that define the latus rectum are $(-4,2),(4,2)$
3. See graph.