Airlines are getting flak for splitting up families so you aren't eligible for the "basic economy" fares when flying with them.

Booking third party might get you a better deal, but there is a big but. The airline can't touch that reservation. So if there is a delay or problem and you need rebooking you will have to do it through the booking sites customer service. Personally, I like being able to book the very next flight from an airline's app while everyone else sits on hold with priceline.