I am looking for a compact CalabiYau 3fold which is fibered by both K3 surface and abelian surface (with possibly singular fibers). Are there any examples?
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 CalabiYau 3fold. With an appendix by Noboru Nakayama. Internat. J. Math. 4 (1993), no. 3, 439465. It is the content of Theorem 4.9 on page 452. It says that there exists a CalabiYau $3$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:
This $X$ is constructed as a certain blowup 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 CalabiYau $3$fold in the most restrictive way, that is $K_X$ is trivial and $X$ is simply connected. 


I believe the following works (although it needs to be doublechecked). For each of $i=1,2$, let $\nu_i:S_i \to P_i$ be the blowing up of a projective plane $P_i$, isomorphic to $\mathbb{P}^2$, at the 9 base points of a general pencil of plane cubics. Let $B_i$, isomorphic to $\mathbb{P}^1$, be the parameter space for this pencil of plane cubics, and let $\pi_i:S_i \to B_i$ be the corresponding elliptic fibration. Thus there is a projection $$\pi_1\times \pi_2: S_1 \times S_2 \to B_1 \times B_1,$$ whose geometric generic fiber is isomorphic to an Abelian surface (in fact $\pi_i$ has a section, so also $\pi_1\times \pi_2$ has a section). By my computation, the dualizing sheaf of $S_1\times S_2$ is the pullback $(\pi_1\times \pi_2)^*[\mathcal{O}_{B_1}(1)\otimes \mathcal{O}_{B_2}(1)]$, where $\mathcal{O}_{B_i}(+1)$ is the unique ample generator of the Picard group of $B_i$ (usual Serre twisting sheaf). Now let $f:B_2\to B_1$ be a general choice of isomorphism. The graph is $g:B_2\to B_1\times B_2$. The normal bundle of the divisor $g(B_2)$ is $[\mathcal{O}_{B_1}(1)\otimes \mathcal{O}_{B_2}(1)]$. Thus, by adjunction, the inverse image $X$ of this divisor under $\pi_1\times \pi_2$ is a divisor in $S_1\times S_2$ that has trivial dualizing sheaf. By Bertini's theorem, for general choice of $f$, $X$ is smooth. Thus $X$ is a smooth, projective $3$fold with trivial dualizing sheaf. Also the fibration $\pi:X\to g(B_2)$ is an Abelian fibration over $B_2$ (I will identify $g(B_2)$ with $B_2$). What about the K3 fibration? Let $x$ be any of the $9$ base points of the pencil of plane cubics on $S_1$. Let $\widetilde{P}_1 \to P_1$ be the blowing up along $x$. Linear projection away from $x$ defines a morphism $\rho':\widetilde{P}^1 \to \Lambda$, where $\Lambda$ is isomorphic to $\mathbb{P}^1$ and $\widetilde{P}^1$ is a $\mathbb{P}^1$bundle over $\Lambda$. Since $S_1$ is the blowing up of $P_1$ along $9$ points that includes $x$, also $S_1$ is a blowing up of $\widetilde{P}_1$, $\mu:S_1\to \widetilde{P}_1$, at the transforms of the remaining $8$ points. Denote by $\rho:S_1 \to \Lambda$ the composition of $\mu$ and $\rho'$. The general fiber of $\rho$ is isomorphic to $\mathbb{P}^1$, but there are $8$ reducible fibers (isomorphic to a union of two copies of $\mathbb{P}^1$ intersecting at a single ordinary double point). Consider the projection $\rho_1:S_1\to S_2 \to \Lambda$ that is the composition of the projection $\text{pr}_1:S_1\times S_2 \to S_1$ with $\rho$. Denote by $\rho_X:X\to \Lambda$ the restriction of $\rho_1$ to $X$. The claim is that a general fiber of $\rho_X$ is an elliptically fibered K3 surface. The fiber $F$ of $\rho$ over a general point $t$ of $\Lambda$ is isomorphic to $\mathbb{P}^1$. Moreover, the projection $\pi_1_F:F\to B_1$ is a degree $2$ cover of $B_1$ branched over $2$ general points. Thus the fiber $X_t$ of $\rho_X$ over $t$ is the fiber product of $\pi_1_F:F\to B_1$ and $f\circ \pi_2:S_2\to B_1$. For general choice of $t$, $X_t$ is a smooth surface. Of course $S_2\to B_1$ has relative canonical bundle isomorphic to the pullback of $\mathcal{O}_{B_1}(+1)$. Thus $X_t \to F$ has relative canonical isomorphic to the pullback of $\mathcal{O}_{B_1}(+1)$, which is the same as the pullback of $\mathcal{O}_F(+2)$. Since $\omega_F$ is isomorphic to $\mathcal{O}_F(2)$, this means that $X_t$ has trivial dualizing sheaf. So $X_t\to F$ is an elliptic fibration over $\mathbb{P}^1$ with trivial dualizing sheaf. Thus $X_t$ is an elliptic K3 surface. Thus $X\to \Lambda$ is a fibration over $\mathbb{P}^1$ by K3 surfaces. 


Let me give a construction which is a bit simpler than the construction suggested by Jason. Let $S$ be an elliptically fibered K3 with a section. Let $C$ be an elliptic curve. Then take just $X = S \times C$. Then the fibers of $X \to S \to P^1$ are products of two elliptic curves, while the fibers of $X \to C$ are K3 surfaces. 

