Quasi-Linear Functions

I understand that quasi-linear functions have a general form

$$U(x_1,x_2,...,x_n,y) = f(x_1,x_2,...,x_n) + y$$

and that for a quasi-linear function, the income effect with respect to the other variables of the function ($$x_1,x_2,...,x_n$$) are all $$0$$, i.e., income has no effect on the consumption of those goods. I also read online that a if a preference is quasi-linear, then indifference curves are parallel.

However, I came across this function, $$U(x,y)=e^xy$$, where the income effect with respect to $$y$$ is $$0$$, when the price of $$y$$ changed. Is this also a quasi-linear function, since IE is $$0$$? Plotting the graph of this function, I see that the indifference curves are actually not parallel.

$U(x,y)$">

The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $$x$$.
The curves you map maintain horizontal distance regardless of $$y$$. That is because the non-linear variable here is $$y$$, not $$x$$. The curves are still shifted, but along the other axis.
Taking the logarithm of the utility function you get $$x + \ln y$$ which is more clearly quasi linear.