Finding the SN explosion time by t-square fitting
I think you misunderstood.

$(y-k) = a (x-h)^2$ is a parabola with its apex at $(h,k)$. Therefore, the formula you mentioned above has $t_0$ as the time at peak, not at the explosion.

For the case of SN, we normally do not know when the explosion happened. Mostly, we have an image before SN detection that can set a constraint for the explosion happening in between the last non-detection to the first detection.

However, most analytical models require the explosion time as the reference, so the parabola fit is applied as an approximation.