$\displaystyle f(x | y) = \frac{f(x, y)}{f_Y(y)}$.
The joint pdf is non-zero for values of X and Y lying in the region of the XY-plane enclosed by x = 0, x = 1, y = 0 and y = x^2. Therefore the marginal density function of Y is
$\displaystyle f_Y(y) = \int_{x = +\sqrt{y}}^{x=1} f(x, y) \, dx$.
My mistake. It should be
$\displaystyle f_Y(y) = \int_{x = +\sqrt{y}}^{x=1} f(x, y) \, dx + \int_{x=-1}^{x = -\sqrt{y}} f(x, y) \, dx = 2 \int_{x = +\sqrt{y}}^{x=1} f(x, y) \, dx $ (since f(x, y) is symmetric in x).
And by now you've realised that it's 1.8 and not 3.3 (see my other mistake in the other post)