I'm not a teacher, so feel free to disregard my opinion.
Pacing is usually used, in this context, when someone is moving back and forth repetitiously. It is usually said when the person doing the pacing is stressed or worried--for example, the classic image of an impatient father pacing back and forth while waiting for his child to be born.
Walking back and forth is also appropriate, but I would say it holds less of the hurried, anxious context pacing does. Whereas saying someone "paced" brings to my mind the image of a harassed-looking individual walking in circles around a room, saying someone was "walking" can allude to any number of situations.
Going back and forth does not tend to apply to people as much as it does objects/ideas. For instance, during a tennis match, spectator's heads often turn to watch the ball going back and forth. Going can also suggest moving frequently between two points, such as buildings at opposite ends of the street.
For that sentence, I would probably choose a different word depending on the situation. If your father was merely moving back and forth, I would say "walking". If your father was waiting on something or for something, I would say "pacing". If your father went to the grocery store, the hardware store, then the grocery store again, I would say he was "going" between the two stores.
I hope that helps, at least a little.
Student or Learner