I haven't seen it myself... my first thought was "maximum connection limit" but traffic wouldn't get backed up THAT far.
Can you tell if the delay is at PostgreSQL's end or PDO's? Log connections and disconnections. Run the PDO connection in isolation so that the only thing the script does is open a connection (which is then immediately closed because the script is finished). Comparing the log timestamps may tell you, between "connection received" "connection authorized" and "disconnection", on which side the delay is.