Impressive query and response. That's how it's done, folks! ;D
As I'm sure you both know, there are sixteen John Petersons in the 1865 Minnesota state census in Goodhue County. The transcriptions on the Minnesota Historical Society website don't show age or ethnicity/place of birth for any of them. There are two John Pedersons, again no age or birth place. Parsing with Petersson and Persson brings up no additional Johns.
Having the parents' names (Peter and Kari) is a huge step forward. Together with the approximate year of immigration, that might be enough to find a Swedish emigration record that fits him. The Sweden board would be the place for help with that. The birth place of his spouses is a possible clue. Immigrants from the same area usually gathered in communities in the USA and thus tended to marry each other. Not true in my family of immigrants - obviously not a hard and fast rule.