I was glad we came to visit the Seaport museum. It offered up enough of a cross-section of maritime history to be surprisingly engrossing. I trained in marine zoology (and my research supervisor was a whale biologist) but still found enough nuggets to satisfy my curiosity for an entire morning.
As the other reviewers noted, the whaling ship (and the various re-enactments / demos) that go along with it are a major highlight. Personally, I found the other segments of the Museum (e.g the cooper, the instrument room, the exhibit on 'streamlining', the life of a sailor during those times) also quite interesting.
The Museum grounds go on for a bit, so allow about 2-3 hours if you plan to do a 'quickie' tour. If you engage with the exhibits, budget around 5 hrs.
Insofar as parking logistics : if you plan to walk into downtown Mystic, park at the South lot. It'll be a 15 min walk . If you find availability only in the North lot, tack on another 10 min.