A plausible hyperbolic link
This link is not hyperbolic. Let $K_1$ and $K_2$ be the link components that cobound the annulus $A$. Let $T$ be the boundary of a regular neighbourhood of $A$ in $S^3$. Let $K_3$ and $K_4$ be the other link components. Let $V$ be the solid torus bounded by $T$ containing $K_3$ and $K_4$.
If $A$ were inessential, then it would compress so that each of $K_1$ and $K_2$ bound disks. But then such a disk would contain a subdisk $D$ which is a compressing disk for $V$. But then $K_3$ and $K_4$ could be isotoped into a ball contained in $V$, and this is impossible. Hence $A$ is essential in the link exterior.
I don't see how SnapPy is telling you that this link is hyperbolic. It does give a nonzero volume, but that is a simplicial volume. It is the hyperbolic volume of the complement of $K_3$ and $K_4$ inside $V$ since $T$ is an essential torus which gives the JSJ decomposition of the link complement. In fact the SnapPy command M.splitting_surfaces() shows that this torus $T$ exists.
As a complement (sorry) to Josh Howie's answer, here is my Snappy session.
In[1]: M = Manifold()
Starting the link editor.
Select Tools->Send to SnapPy to load the link complement.
New triangulation received from PLink!
In[2]: M.volume()
Out[2]: 7.327724753
In[3]: M.solution_type()
Out[3]: 'contains degenerate tetrahedra'
Snappy is "guessing" that the manifold is not hyperbolic, but does have a hyperbolic piece in its JSJ decomposition. (There is a lot of accuracy in the volume computation!) As Josh Howie points out, you can actually use Snappy (running under Sage) to prove that the manifold is not hyperbolic, and that that piece is a Borromean rings complement. (The .identify() method is your friend here.)
Added later: So I got snappy running under sage. Here is that session.
sage: %gui tk
sage: M = snappy.Manifold()
Starting the link editor.
Select Tools->Send to SnapPy to load the link complement.
New triangulation received from PLink!
sage: M.volume()
7.32772475341777
sage: M.solution_type()
'contains degenerate tetrahedra'
sage: M.splitting_surfaces()
[Orientable two-sided with euler = 0]
sage: M.split(0)
[unnamed link.a(0,0)(0,0)(0,0), unnamed link.b(0,0)(0,0)(0,0)]
sage: A, B = M.split(0)
sage: A.identify()
[t12067(0,0)(0,0)(0,0),
6^3_2(0,0)(0,0)(0,0),
L6a4(0,0)(0,0)(0,0),
ooct02_00005(0,0)(0,0)(0,0)]
sage: B.identify()
[6^3_3(0,0)(0,0)(0,0)]
sage: B.volume()
-1.45994327738208e-14
sage: A.volume()
7.32772475341775
and we are done. Snappy identified the two pieces (after splitting) as link complements. Thus the splitting torus is incompressible. So the manifold A (which is the Borromean rings complement) is a geometric piece of the JSJ decomposition. The other piece, a three-times punctured sphere crossed with the circle, is Seifert fibered and so is the other piece of the JSJ decomposition.