TITLE: If it quacks like a conifold resolution and it waddles like a conifold resolution, $\ldots$
QUESTION [8 upvotes]: 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?

REPLY [5 votes]: Take a conifold resolution $\tilde{X}$ and then "impose a cusp" on its exceptional fiber (in the vertical tangent direction). In other words, in a local chart replace $Spec(A)$ by the spectrum of the subring 
$$
A' := \{a \in A \mid \partial(a) \in \mathfrak{m} \},
$$ 
where $\mathfrak{m}$ is the ideal of a point on the exceptional fiber and $\partial$ is a derivation tangent to exceptional fiber. Define $Y$ by gluing $Spec(A')$ with the rest of $\tilde{X}$. This is a counterexample to your question.