Finding the SN explosion time by t-square fitting In a couple of papers I've read that it's usual practice to fit early rise of supernova luminosity curve by $t^2$. Let me describe how I see it:

I take the early fast rising part of luminosity curve and then fit them to a curve:

$L = a\cdot\left( t - t_0 \right) ^ 2 + L_0,$

Where $t_0$ is the moment of explosion.

Is this approach right?
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.