Both, according to definitions.
Let consider the two balls as the components of the system. Well, unless we consider the wire as a component. In any case it is obvious the two spheres are connected in series.
But they are also connected in parallel as a trivial or degenerate case of the definition of parallel circuit.
By definition the terms series and parallel relate to a circuit.
A circuit composed which is solely of components connected in series is known as a series circuit. In such a circuit the current flowing through each component of circuit is same.
Similarly, a circuit in which components are connected completely in parallel is known as a parallel circuit. The current flowing through the components gets divided as ascertained by applicable laws. We must remember that in a parallel circuit the connecting wires are in series to the device these connect.
Now coming back to the question. The statement is incomplete. It has not been qualified if the connecting wire is conducting or non-conducting.
Assuming the intention that it is a conducting wire.
If the spherical conductors are originally at the same potential, there is no flow of current in any of the components of assumed circuit at any point of time.
In case these are at different potential, there is going to be redistribution of charges.
The questions arise:
1. Whether current flows the conducting wire?
The answer is : Yes.
2. Whether current flows through the conductors?
The answer as I understand is : No.
*If you think otherwise, please edit the answer appropriately along with Explanation.