The time and place of "The Father" are not given explicitly in the story. Bjørnstjerne Bjørnson most likely intentionally left the time and place unspecified in order to stress the universality of the tale. However, we can make a few educated guesses from what is provided to us. We can get a vague idea of the setting from looking closely at the text.
Most of the action takes place at the parish church. Four times Thord comes to the church to make arrangements on behalf of his son. The only scene to take place elsewhere is the one on the lake, where Thord's son drowns. It is likely that the church and the lake are in Norway, the author's homeland, or possibly Sweden, a country he had close ties to. The characters' names and actions are purely Scandinavian, so it is likely that the story is meant to take place in one of these two countries.
"The Father" was written in 1860, and there is no reason to believe that it takes place at any other time. However, there are no clues in the text to indicate any particular time or era of the story.
The fact that the time and place of the story are left out serves to make "The Father" relatable to anyone. The themes of this work are not confined to any one place, time, or people. That the story likely takes place in a location familiar to the author (and during his time period) has more to do with Bjørnson trying to make the parable relatable to all his contemporary readers than it does with the plot itself.