I’ve asked several people and they seem to have different opinions. Even native speakers seem to disagree on the nuance of this sentence:
息子が子供の時、よく川で遊んだ
Who played in the river in this sentence? Was it the son or the ‘I’ (the parent)?
My teacher said that it should be read as 息子が子供のとき、(私は)よく川で遊んだ。 So the sentence means “when my son was a child, I often played in the river”.
However, some people seem to think ‘the son’ is implied as the ‘topic’. But in that case, my argument is that it should be 子供のとき、息子*は*よく川で遊んだ。 that is, the topic should be mentioned specifically in the second clause, and in this case omitted from the first one because it’s the same (you don’t need say 息子が子供の時、息子はよく川で遊んだ)
Some other people say it’s ambiguous and that maybe they played together…
Basically as I understand it the は part illustrates ‘who played in the river’, and should always be thought of as 私 if omitted, and the が part illustrates who was a child, and defaults to the target of the は if omited.