DEAR EXPERTS ....HERE ALONGY AXIS ARE WE NEGLECTING 1/₂ML².....IF NOT THE ANSWER OF SUMMATION SHOULD BE 3/₄ML²..

Dear student, 

For the rod along (y-axis) parallel to the axis of rotation,
$I_2=M{r}^2=M{\left(\frac{L}{2}\right)}^2=\frac{M{L}^2}{4}$
There is no need to add any extra term in this.
The reason is MOI of rod along its own axis is zero.

Regards.