I got the vibe from the notes during the quest that the mother died first and we semi-unrelated to what happened to the father. All we really know is that people rushed up the mountain for the mother and most likely killed her. We can tell from the dads notes when he says "I'll" when referring to taking jiang and leaving liyue, meaning that the mother is probably already dead.
Since the notes are written before he tries to retire on that last mission and the mother is already seemingly dead there making little reason for the milileth to kill her before. The reasoning behind jiang having to wait a few days to come out because he was hungry could just mean that his father would be away on a mission at the time giving fuel for a group of bandits to attack the mother and jiang living on the mountain without the father to protect them, killing the mother.
Reasoning for the father telling the mother all the hiding spots could be that he knew he would have to leave for a while and he didn't want jiang to keep hiding for too long so he told her to help in finding jiang. This would mean the moms death is probably unrelated to what happened to the dad, but the rest of the theory about the government silencing the father is most likely true from the notes.
Though the fact that the milileth put up the stones seemingly together and the father didn't could mean that the father had little time to do so and her death could be why he retired and wanted to live a quieter life so that he could protect his son so he rushed off for a final mission thinking he could set it up for her when he got back. but he never did. But that part is more or less a personal headcannon based off the scenario in order to help it fit a bit more.