I don't think you need to worry - he's trying to manage his pain without having to go to the Doctors. Paracetamol is quite a weak analgesic (pain killer) and therefore more are needed for the desired effect. If he is overdosing, which I imagine he is (unintentionally) then there can be liver and kidney problems.
He needs to go see the GP for some stronger stuff, which will probably be co-codamol or codeine. These are stronger analgesics and he shouldn't need to take as many. But mostly they are prescription only.
As far as the tiredness goes - your Dad has had surgery, cancer, and has been taking paracetamol to excess. Unwanted effects of these results in tiredness, plus he's probably anxious about his medication and the pain.
Just give him some reassurance, and approach it head on with him - tell him you're concerned about how much medicine he's taking, when he could be taking something more effective at lower doses. You can't force him to the Drs but he should decide to go himself. He should be ok in the meantime, just support him and continue to do a wonderful job, which I'm sure you are.