  The complexity of the maximum common connected subgraph problem in partial
$k$-trees is still not fully understood. Polynomial-time solutions are known
for degree-bounded outerplanar graphs, a subclass of the partial $2$-trees. On
the other hand, the problem is known to be ${\bf NP}$-hard in vertex-labeled
partial $11$-trees of bounded degree. We consider series-parallel graphs, i.e.,
partial $2$-trees. We show that the problem remains ${\bf NP}$-hard in
biconnected series-parallel graphs with all but one vertex of degree $3$ or
less. A positive complexity result is presented for a related problem of high
practical relevance which asks for a maximum common connected subgraph that
preserves blocks and bridges of the input graphs. We present a polynomial time
algorithm for this problem in series-parallel graphs, which utilizes a
combination of BC- and SP-tree data structures to decompose both graphs.
