The question is as follows:
Find the graph of the parametric equations defined by
$$ x(\theta) = \sin \theta \\ y(\theta) = 3 - 2\cos(2\theta) $$
We are supposed to use the identity that
$\sin^2\theta \ + \cos^2\theta\ = 1$
However, that identity requires that sin and cos both have the same theta, and in this instance they are different.
$$x(\theta) = \sin \theta \\ y(\theta) = 3 - 2\cos(2\theta)$$
Note that $$ \cos (2\theta) = 1-2\sin ^2 (\theta)$$
The expression for $y(\theta )$ simplifies to $$y(\theta) = 3 - 2\cos(2\theta)=1+4\sin ^2 (\theta) = 1+4 x^2$$
Thus your parabola is simply $y=1+4x^2$ where, $-1\le x\le 1$