Actually, the wording implies a single adult. But that's generally never enforced, and two parents are often allowed to board with one child. I also believe that the age cutoff is fairly arbitrary -- in fact, Southwest raised the age to 6 from 4 in 2015. I suspect you'll have no issues boarding with both kids during Family Boarding (although, if you are stopped by the OPs agent, be prepared to wait and board at your assigned positions.)
From the website:
An adult traveling with a child six years old or younger may board during Family Boarding, which occurs after the “A” group has boarded and before the “B” group begins boarding. If the child and the adult are both holding an “A” boarding pass, they should board in their assigned boarding position.
If you're still concerned, you could 1) check in at as close to exactly t-24 hours as possible, or 2) purchase EarlyBird Check-In for you and the 8 year old. At $15 per person, one-way, that would check you in automatically and give you a "better" position than t-24 hours check in. Positions are based on the order of purchase, so if your flight is coming up soon I wouldn't recommend this option. The farther in advance EBCI is purchased, the better.
My recommendation would be to try Family Boarding. If I were you, I'd be more concerned about being questioned about your 23 month old still being a lap child (assuming you haven't bout them a seat). Be prepared to show his/her birth certificate at the desk when you check in at the airport, in order to receive his/her boarding document.
