Finding 2 * J class on any given flight is always going to be easier than finding 4 on the same flight as limited classic reward seats are available on each flight. So one option is considering 1 parent and 1 child on flight and the other parent and child on different flights if needed. Or parents in J and children in PE or Y if old enough to be in separate cabin.
QF released a bunch of reward seats 2 weeks ago, I noticed that irrespective of which route i queried (US, UK, EU) i was almost always offered routing via HKG on CX or via Tokyo on JL. So Asian destinations seem to have good availablity, as does taking the long way to/from the US. Reward J seats on QF metal are almost alsways rarer than on One World partners.
Best availablity is always outside of school holidays and flying on days that are less popular for business travel. I notice QF to LHR has best availablity on Tuesday departures.
Your status determines how far out you can see seats. And you may need to get creative chosing non direct options to find the number of seats you need.