Applying for a UK visa when family members are traveling different portions of the trip

what's best way to apply

"Best" does not have meaning in the context you have written about. There's the only way to apply. And once that is nailed down, you can add some 'best practices' on to it...

• Have the daughter prepare a single invitation letter addressed to the Entry Clearance Officer at the decision-making hub in Abu Dhabi, add the proposed travel arrangements and explain why they are necessary; send the letter to the parents;
• Have the daughter collect a single set of bank statements, tenancy contract or mortgage, job contract, send it to the parents;
• Have each of the travellers (including parents) prepare the same bundle of evidence;
• Have each of the travellers create an account a Apply for a UK Visa and complete the application;
• Report to the Visa Application Centre in either Dubai or Abu Dhabi for biometric enrolment; and
• Await results.

They are going to wonder why 4 people in the UAE need 4 open visas to the UK at the same time.

Since the proposal involves having the two UAE sons travelling to the UK as escorts, there's an opportunity to substantially reduce the overall risk by having the UK-based daughter provide all the escorting. I assume the difference in airfares is minimal since either way entails two round trips.

Other notes...

• Bookings, tickets, and confirmations are not necessary for visit visas.
• The itinerary should include transport to and from the airport, especially if the parents are not comfortable with spoken English.
• Tenancy agreements (or mortgage info) will be key, make sure it's up-to-date.
• If the UK-based daughter has visited the parents in the UAE recently, it can be worthwhile emphasising this in the application, if not then consider offering a really solid reason why not.