How do you identify transformations in parent functions given #y = 2(x-3)^2#?

Look at the $a$-value and $h$-value (and the $k$-value): the $a$-value transforms the parabola. The $h$ and $k$-value translates the parabola.
In this case, the transformed parabola has a vertical stretch by a factor of $2$ and is translated $3$ units to the right. The parent function if $y=x^2$, which looks like this:
graph{x^2 [-10, 10, -5, 5]}
The transformed function, $y=2(x-3)^2$ is a lot more simpler to determine its transformation because it is given in vertex form.
There are two main things being done to the parabola.

The $a$-value - having a value greater than $1$ as the $a$-value indicates a vertical stretch by a factor of whatever was used. In this case, $2$.
The $h$-value translates the parabola to the left or right. It is determined by isolating the $x$-value in the bracket (AND BRACKET ONLY). In this case, the parabola is moved $3$ units to the right.

It looks like this:
graph{2(x-3)^2 [-10, 10, -5, 5]}
Hope this helps