Hi Greg!
Records of births, deaths and marriages were kept only on a local level in the parishes until 1860.
There are gaps, but only local ones (caused by the records being destroyed in church-fires or similar).
With few exceptions the records are fairly complete from the late 17th century and onwards. So I believe it is very likely that your relative is to be found in the archives - the question is where
.
The indexes of these records are, unfortunately, not at all complete so far.
Swedish sailors were supposed to be registered at a "sjömanshus" (literally "sailors house"). Some of sjömanshusen kept tidy records with alphabetic indexes, while others, in my experience, are more chaotic.
The problem here is twofold: We do not know which sjömanshus he was registered at, and we do not know what name he used in Sweden (probably not Brown).
http://marinersandships.com.au/1854/12/033lad.htmIs this the record from your relatives arrival? The man called Broom? Which sounds no more swedish than Brown!
The next step could perhaps be to check if there are any records of the crew from the port of departure (Plymouth).
Maybe this is the right place to look for such departure records, I am not sure.
https://discovery.nationalarchives.gov.uk/details/r/C4068301These records have not yet been digitised, but they have been transfered to microfilm by the mormons.
https://www.familysearch.org/search/catalog/55017?availability=Family%20History%20LibraryIn summary, I think it would be necessary to find out more about Peter Browns background before searching the Swedish archives; until we know more specifically where he was born or lived in Sweden, or what name he used, finding him would be very difficult.
Erik