This is only a partial answer.

Threefolds $V$ with an isolated singular point $p$ such that there is a small resolution $\pi \colon X \to V$ with exceptional set isomorphic to $\mathbb{P}^1$ were investigated by H. B. Laufer in the paper

On $\mathbb{CP}^1$ as an exceptional set, Recent Developments in Several Complex Variables 100, Annals of Mathematics Studies (1981).

Laufer proves the following

Theorem. Let $V$ be an analytic space of dimension $n \geq 3$ with an isolated singularity at $p$. Suppose that there exists a non-zero holomorphic form $\omega$ on $V - \{p\}$ (i.e., that $p$ is a Gorenstein singularity). Assume moreover that there exists a small resolution $\pi \colon X \to V$ with irreducible exceptional locus $C := \pi^{-1}(p)$.

 

Then $C$ is isomorphic to $\mathbb{P}^1$, $n=3$ and the possible bidegrees for the normal bundle $N_{C/X}$ are $(-1, \, -1)$, $(-2, \, 0)$ and $(-3, \, 1)$.

 

In the $(-2, \, 0)$ case the local form for $V$ around $p$ is given by a hypersurface in $\mathbb{C}^4$ of equation $$x^2+y^2+z^2+w^{2k}=0,$$ with $k \geq 2$ (whereas for $k=1$ we have normal bundle of type $(-1, -1)$).

I did not have the time to check if for some of these example the small resolution $X$ is a Calabi Yau threefold. If so, this would provide an affermative answer to your question.

This is only a partial answer.

Threefolds $V$ with an isolated singular point $p$ such that there is a small resolution $\pi \colon X \to V$ with exceptional set isomorphic to $\mathbb{P}^1$ were investigated by H. B. Laufer in the paper

On $\mathbb{CP}^1$ as an exceptional set, Recent Developments in Several Complex Variables 100, Annals of Mathematics Studies (1981).

Laufer proves the following

Theorem. Let $V$ be an analytic space of dimension $n \geq 3$ with an isolated singularity at $p$. Suppose that there exists a non-zero holomorphic form $\omega$ on $V - \{p\}$ (i.e., that $p$ is a Gorenstein singularity). Assume moreover that there exists a small resolution $\pi \colon X \to V$ with irreducible exceptional locus $C := \pi^{-1}(p)$.

Then $C$ is isomorphic to $\mathbb{P}^1$, $n=3$ and the possible bidegrees for the normal bundle $N_{C/X}$ are $(-1, \, -1)$, $(-2, \, 0)$ and $(-3, \, 1)$.

In the $(-2, \, 0)$ case the local form for $V$ around $p$ is given by a hypersurface in $\mathbb{C}^4$ of equation $$x^2+y^2+z^2+w^{2k}=0,$$ with $k \geq 2$ (whereas for $k=1$ we have normal bundle of type $(-1, -1)$).

I did not have the time to check if for some of these example the small resolution $X$ is a Calabi Yau threefold. If so, this would provide an affermative answer to your question.