You can find such an example (which gives you much more indeed) in the following paper:

Oguiso, Keiji. On algebraic fiber space structures on a Calabi-Yau 3-fold. With an appendix by Noboru Nakayama. Internat. J. Math. 4 (1993), no. 3, 439-465.

It is the content of Theorem 4.9 on page 452. It says that there exists a Calabi-Yau $3$-fold which admits a Type $II_0$ fibration and infinitely many different fibrations of each of the three types $I_+$, $I_0$ and $II_+$.

In his notation, roughly speaking, these fibration types are respectively:

• $I_0$, fibration on $\mathbb P^1$, general fiber an abelian surface.
• $I_+$, fibration on $\mathbb P^1$, general fiber a $K3$ surface.
• $II_0$ and $II_+$, fibrations on a rational surface, general fiber an elliptic curves (they are distinguished by the intersection of the second Chern class of $X$ with the class of the divisor defining the fibration).

This $X$ is constructed as a certain blow-up of the quotient $\overline X$ of $Y=S\times G$ by a canonically defined involution, where $G$ is an elliptic curve and $S$ a $K3$ surface with an involution whose fixed points locus is a union of rational curves. I refer to the original paper for more details!

Finally, notice that for Oguiso $X$ is a Calabi-Yau $3$-fold in the most restrictive way, that is $K_X$ is trivial and $X$ is simply connected.

You can find such an example (which gives you much more indeed) in the following paper:

Oguiso, Keiji. On algebraic fiber space structures on a Calabi-Yau 3-fold. With an appendix by Noboru Nakayama. Internat. J. Math. 4 (1993), no. 3, 439-465.

It is the content of Theorem 4.9 on page 452. It says that there exists a Calabi-Yau $3$-fold which admits a Type $II_0$ fibration and infinitely many different fibrations of each of the three types $I_+$, $I_0$ and $II_+$.

In his notation, roughly speaking, these fibration types are respectively:

• $I_0$, fibration on $\mathbb P^1$, general fiber an abelian surface.
• $I_+$, fibration on $\mathbb P^1$, general fiber a $K3$ surface.
• $II_0$ and $II_+$, fibrations on a rational surface, general fiber an elliptic curve (they are distinguished by the intersection of the second Chern class of $X$ with the class of the divisor defining the fibration).

This $X$ is constructed as a certain blow-up of the quotient $\overline X$ of $Y=S\times G$ by a canonically defined involution, where $G$ is an elliptic curve and $S$ a $K3$ surface with an involution whose fixed points locus is a union of rational curves. I refer to the original paper for more details!

Finally, notice that for Oguiso $X$ is a Calabi-Yau $3$-fold in the most restrictive way, that is $K_X$ is trivial and $X$ is simply connected.