Suppose that $X$ is a projective threefold with at worst conifold singularities and suppose $\omega_X$ trivial. Suppose $Y$ is a projective variety with a birational morphism $f: Y\to X$ which is an isomorphism away from the conifold points and such that $f^{-1}(p) = \mathbb{P}^1$ for each conifold point $p \in X$. Can I conclude that $Y$ is smooth? i.e. that $f:Y\to X$ is a conifold resolution?

This seems too good to be true, but I was unable to come up with a counterexample and it would be really useful (to me at least) if it were true.

Suppose that $X$ is a projective threefold with at worst conifold singularities and suppose $\omega_X$ trivial. Suppose $Y$ is a projective variety with a birational morphism $f: Y\to X$ which is an isomorphism away from the conifold points and such that $f^{-1}(p) = \mathbb{P}^1$ for each conifold point $p \in X$. Can I conclude that $Y$ is smooth? i.e. that $f:Y\to X$ is a conifold resolution?

This seems too good to be true, but I was unable to come up with a counterexample and it would be really useful (to me at least) if it were true.

Edit: In light of Sasha/Jason's counterexample I would like to impose a normality condition. The most useful version to me would be to let $\bar{Y} \to Y $ be the normalization of $Y$, and ask: is $\bar{Y} \to X$ a conifold resolution? Alternatively, is it true if I assume that $Y$ is normal to begin with?