Some have answered your other questions, as to this specific aspects of lounges: Oneworld Sapphire will get you into the following HKG lounges
* Cathay Pacific's The Deck
* Cathay Pacific's The Wing Business
* Cathay Pacific's The Pier Business
* Qantas Hong Kong Lounge

Of these, only The Deck, The Pier Business and the Qantas Hong Kong Lounge have showers. I prefer the showers at The Pier and The Deck, as they are very well-designed and finished. [Edit: that doesn't mean the Qantas Lounge showers are shabby in any way, they're also fine and spacious, I just really like the way Ilse Crawford designed the showers at The Pier, I find for instance that they don't steam up, the water from the shower doesn't splash all over the floor, and by memory there's ample room for one's own toiletries etc on the shelf. These things all make a big difference to me!)]

The Pier Business (and no other lounge on the list above) has a nap area tucked way down the end of the lounge (where it's quietest, which of course makes perfect sense) – PR photo attached for your reference.



However, I personally find these are not super-comfortable for sleeping or even napping due to their relatively short length and the angled 'pillow/headrest' at the top, which can't be removed. I simply can't relax with my upper back and beck in that position.

As to which lounge/s you visit – with your Dad in tow you might not wish to go lounge-hopping, so it'll come down to the departure gate for your onwards flight to LAX.

The Deck, The Wing and the Qantas lounge are all at the 'top' of the airport: if you go through the airport immigration area nearest the CX checkin desks, The Wing is on your left (around gates 1-4), with The Deck and the Qantas Lounge at the other side, around gate 7.

Of the three, I'd actually lean towards the Qantas Lounge as first pick, although it's very handy to The Deck if you also want to pop in there for a visit to The Noodle Bar. (You can actually take the rear exit from the Qantas Lounge and walk straight across to The Deck, although to re-enter the Qantas Lounge you'll have to go downstairs and back up again).




The Deck might also be nice for you and your Dad as a change of scenery, especially for sitting out on the terrace area with a view over the tarmac and runway.

That said: if your LAX flight leaves from a gate around the low-30s or higher, you may as well go to The Pier Business, which is down that end of the airport anyway. You can catch the 'automated people mover' down to that end of the terminal to make things easier on your Dad, if that's a concern.