Does the FAA allow a jet bridge to approach an aircraft that has engine #2 running?
On a two engine airplane, Engine #2 will be on the opposite side from the jetway, and keeping the engine running while the bridge is connected on the other side is not an issue.
(see related question: How are engines numbered?)
On a 4 engine airplane, Engine #2 will be directly behind the jet bridge, and should be shut down before the bridge is extended. This is for the safety of the engine, to avoid ingesting any debris from the jet bridge, or debris dropped by people, rather than for the safety of the people (who are unlikely to be sucked in by an engine at low power).