Hi Monica,
Have you checked out the Swedish-American church record in the areas in which they lived? If the parents joined the church, there will be information on when/where they were born in Sweden and when they came to the U.S., etc. I don't see anything at the Swenson Center for Bay City but these records have been filmed for Pierce County, Wisconsin. Maybe they attended one of these churches or at least took Amanda there for her baptism.
WI PIERCE ELLSWORTH BETHEL MISS COVENANT CHURCH C 1880 S-43
MINS1878-96, 1937-65
WI PIERCE ELLSWORTH ZION COVENANT CHURCH C 1890 S-33
HIST-1965;MINS-1934;MEMB'16-58
WI PIERCE HAGER CITY SVEA LUTHERAN CHURCH L 1875 S-29
MINS1875-1930;MEMB1875-1958;M/A'75-58
In case the Red Wing birth location is more likely to be correct, try these. (The Swenson Center has plenty more for other parts of Goodhue county too.)
MN GOODHUE RED WING FIRST BAPTIST CHURCH B 1892 B-49
HIST1892-1942
MN GOODHUE RED WING FIRST COVENANT CHURCH C 1874 S-444
HIST1874-1974;MINS1873-1893;MEMB'75-47
MN GOODHUE RED WING FIRST LUTHERAN CHURCH L 1855 E-16
MINS1855-1935;MEMB1855-1968
MN GOODHUE RED WING FIRST LUTHERAN CHURCH L 1855 E-17
M/A1855-1925
Both Minnesota and Wisconsin had STATE censuses, usually in years ending with 5, in addition to federal censuses, in years ending with 0. It seems quite strange that they can't be found in ANY of those censuses. It is perhaps possible to be missed in one census but if they are missed in several censuses, something is wrong. I suspect they either didn't live there or you are relying on exact spellings, and those names you have are Americanized. (Swanson would have been Svensson in Sweden, and the first stage of Americanization would probably have been to use Svenson or Swenson. Use the Soundex search and not a search with exact spelling.
Births at sea can be found at the end of the ship's manifest, once you find the ship's manifest, of course.
There are many sources which are too lengthy to mention and which you have apparently not used. These pages tell about them.
http://www.rootsweb.com/~rwguide/ http://www.rootsweb.com/~newbie/ http://rootsweb.com/~genepool/sources.htm Most of those will not be found online. You need to go to archives and libraries, write letters, send for documents, order films from a Family History Center, go to a NARA facility, etc.
This is the url for the Swenson Swedish Immigration Research Center. (mentioned earlier). Its records are extensive but they aren't online.
http://www.augustana.edu/administration/SWENSON/ It is MUCH easier to do genealogy if you start with yourself and work backwards one generation at a time. Start with the known and slowly work back towards the unknown, going from certainty to certainty. The hardest way to do genealogy is to skip those steps. It may seem faster to skip those steps but when you hit a brickwall, as you have done, you need to gather more clues in the generations closer to you and then your progress is stalled. It is truly faster to follow the methodical path of starting with yourself and working backwards. (I've tried both ways, so I know from experience what each way is like.)
Join genealogical societies. I learned much about how to research from those. Join one in your area so you will have support in the sometimes difficult stage of getting started, but also join societies in the areas you are researching, so you can more easily learn about local resources and so you can receive local guidance.
Good luck!
Judy