William (or more likely, Wilhelm/Vilhelm) was born early enough that his name in Sweden would have been Hansson (son of Hans), not Persson (son of Per). He was the grandson of Per something, not the son of Per. No matter what name he used afterwards (assuming he emigrated), he most likely used his patronymic name in Sweden.
At that time period, Christina would have avoided the "son" ending, most likely. She was female, so she probably used the daughter (dotter) ending to her name.
Ystad is a kommun (municipality/township) so it would consist of a number of parishes.
Ystads fögderi Baldringe Balkåkra Bjäresjö Borrie Bromma Glemminge Hedeskoga Herrestad, se Stora Herrestad Högestad Hörup Ingelstorp Kloster, se Ystads Sankt Petri* Löderup Sjörup Skårby Snårestad Stora Herrestad Stora Köpinge Sövestad Valleberga Västra Köpinge, se Stora Köpinge Ystad Ystads garnisonsförsamling* Ystads Sankta Maria, se Ystad Ystads Sankt Petri* Ystads stadsförsamling, se Ystad Öja
Unfortunately, that name is *******exceedingly******* common and without a definite birthdate (day/month/year) and parish, it would be hard to find him.
I suggest using the Swenson Swedish Immigration Research Center to try to find more clues. Also check out the immigration year in the 1900-1910-1920-1930 U.S. Federal Census and the month/year of birth listed in the 1900 U.S. Federal Census.
Also try the Sweden forum. Type Sweden into the "Jump to Forum" space at the top right of this page. Perhaps someone there has a database which can help. No one person has access to every database and although I own quite a few, I certainly don't own everything that is available.