I think you mean that your DC1 will be staying in the UK and each parent is going on one week's holiday, Mum + other DCs in week 1 and Dad + other DCs in week 2, to the same place?
In that case it might get complicated if you book online because the booking form will probably assume that everyone who flies out at the start of week 1 will fly home together at the end of that week, whereas of course that will be Mum only. If you booked a package for Mum + other DCs for week 1 and Dad + other DCs for week 2, you would then be paying unnecessarily for an extra return and then outward flight for the DCs (which they wouldn't take).
You could also book two weeks for one adult and the DCs, and then try to find a way to have the name of the lead passenger change halfway through. The problem is that while the hotel might accept this, the airline is unlikely to.
Or you could book a week for Mum, a week for Dad, and two weeks for the DCs. But this will probably not be accepted because on each booking you will need a lead passenger who is 18 or over.
I think you are going to need to talk to a real human travel agent to see what they can magic up. And of course a package holiday is just a hotel (which you can book from their site), some flights, and a transfer, so you could also DIY.