The two people have shared custody of their son, so he spends time at both of their houses.

To pick up is a phrasal verb that has various meanings.  Here it means to get into your car, drive somewhere where a specific person is, that person gets into your car with you, and you drive away.

I'm going to the zoo today, but I have to pick my nephews up on the way there.

Tom would like to come to the party tonight, but he needs a ride.  Would you mind going and picking him up?

"Picked up" means you went to get something and it wasn't a lot of trouble to do so.

He "picked up" Andrew, so he went to Kim's house and took Andrew back to his own house for a little while. According to the story Tru and Kim had a very good relationship after they were divorced, so Tru didn't just "pick up" his son and leave right away, instead they sat together for awhile and talked pleasantly and then they left.

To "pick" something gives the impression that it is very light and easy. I will "pick" a flower (or he "picked" a flower if it is in the past), it means that you just reach down and pull the flower up without any effort. Perhaps that's why "picked up" refers to easy little things that we do during the day.

"I picked up my phone and called my friend, I told him I would pick him up in an hour and we would go to the store to pick up his new television" - this means everything is being done in a very causal way without any trouble or too much effort.

"I picked up my phone and called my friend, I told him I would drag myself over to his house in an hour so we could go to the store and haul his new giant television back to his house" - means this person called his friend without too much effort, but that he thought the other tasks were bothersome and annoying and that they required a lot of effort. Words like "haul" and "drag" means they were heavy and required a lot of effort. "Pick" means it was something light and easy.