Complexity of the directed Steiner tree problem on special graph classes
This manuscript seems to prove exactly that. (It doesn't; the complexity parameter is $|T|$, not the treewidth.)

In general, most NP-hard optimisation problems have polynomial-time algorithms when the input is restricted to bounded-treewidth graphs. These algorithms use the rather well-known tree-decomposition machinery, which is also used in the linked paper.

EDIT: The undirected Steiner tree is, on the other hand, known to be fixed-parameter tractable with regards to parameter $w$ the treewidth of the underlying graph. My suggestion would be to try and adapt this algorithm to the directed case, which would in particular give a polynomial time algorithm for series-parallel graphs.