I wonder why so many people don't use the Sweden forum. It is in English. Type Sweden into the "Jump to Forum" space at the top right of this page. Include a link to this thread so that they don't waste their time providing the same information I provided. You certainly would like new information and not just a repeat of the following.
Here are two possibilities for Karl Fagerström. Hopefully you have a more complete birthdate to compare.
The Emibas CD is an emigration database made from registrations in the Swedish parish to go to a foreign country. It is about 75% complete.
Fagerström, Karl Fritiof Dräng = farmhand. (unmarried man)
b. 12/1/1866 in Längjum, Skaraborgs län (Västergötland)
Emigrated 4/1/1884 from Tomten, Längjum, Skaraborgs län (Västergötland) to Amerika
Source: Household Examination Roll, p. 107
Emibas migration file ID: Längjum R 1884 003
Fagerström, Karl Johan K Yngling = youth (unmarried man)
b. 2/11/1866 in Smedby, Kalmar län (Öland)
Emigrated 7/6/1883 from Stora Smedby, Smedby, Kalmar län (Öland) to Nordamerika
Source: Household Examination Roll, p. 60
Emibas migration file ID: Smedby H 1883 009
Most of our Swedes left Sweden with just their patronymic last name (ending with son or dotter) and not a surname such as Fagerström, so yours might have done that too. If so, far more information is needed to help identify your Carl/Karl from all the other Carl/Karl hits with no last name born about 1866 and emigrating in an unknown year.
Emihamn was made from registrations with Swedish port police before leaving Sweden.
This man on Emihamn looks like Emibas hit # 1, but the parish name from which he departed doesn't match. Larv parish is the mother parish in the Larv, Länjum, and Tråvad pastorat, so I think the moving out certificate he was issued was probably issued by the mother parish.
Note the destination! Now you need to try to find a complete birthdate in U.S. records to match the birthdate found on Emibas.
First name: CARL Last name: FAGERSTRÖM Age: 17Gender: M Parish: LARVCounty: R (Skaraborg) Port: GÖTEBORG Date: 1884 04 04 (April 4) Destination: ROCKFORD Fellows: NEJ Source: 24:282:31922
This man matches the second man found on the Emibas CD.
First name: KARL Last name: FAGERSTRÖM Age: 17Gender: M Parish: SMEDBYCounty: Port: KALMAR Date: 1883 07 11 (July 11) Destination: NEW YORK Fellows: NEJ Source: 3805a:6:7221
********************************* You didn't tell us when Amanda Maria Sparf was born or when she emigrated. Could she be this person found on the Emibas CD?
Andersdotter Sparf, Amanda Maria Unmarried woman
b. 7/25/1871 in Ölmstad, Jönköpings län (Småland)
Emigrated 5/22/1882 from Hunseberg Vestreg., Ölmstad, Jönköpings län (Småland) to Nordamerika
Source: Household Examination Roll, p. 106
Emibas migration file ID: Ölmstad F 1882 011
I don't see her on the Emihamn database so probably she left Sweden from a port in another country such as Denmark, Norway, or Germany. *********************** Rockford was and is a very Swedish community. I suggest that you contact the Swenson Swedish Immigration Research Center. The Swenson Center has a website which you can google, but its extensive collection of records is NOT online. We need to go there to research or pay them to research for us. They have materials which are difficult or even impossible to find elsewhere, and many of us have found the parish of birth in Sweden and much about their early lives in the U.S. by using their services.