We flew out of Dulles on SW last month and they were enforcing the child plus one parent rule. (Also the blue sleeve plus one). There was a family with a baby and only the mom and baby were allowed to board. Originally the entire clan was in the line. I saw the FA go over and the next thing I saw was everyone went back to their seat except mom and baby. I'm guessing this group expected to board with the family because they were all the last to board, so I expect they had C's. Didn't really matter as mom saved seats for 2 of them, and they were traveling with a wheelchair person, so only the wheelchair person plus one extra was allowed to pre-board. Between the 3 of them, they saved the first 3 rows on the flight, so the rest of the party still got to sit with them.
OP, I certainly would get those boarding passes just in case the one-parent rule is being enforced that day.
This is from SW website:
An adult traveling with a child four years old or younger will board between the “A” and “B” boarding groups. If you have an “A” boarding pass, please feel free to board with that boarding group. With an all-jet fleet outfitted with comfortable, leather seats, our families with small children are easily accommodated together.