I'm surprised too that it's Braniff, because their own timetables of the era gave no indication of whether flights were coach only; you are just left to assume they all are two class. Here's a page from their April 1982 schedule, for example
http://www.departedflights.com/BN042582p6.html
Now the OAG was always precise about class of service for every flight, so their data would be different. By the way, you don't think that the OAG could have had a typo, do you ? I've come across several over time.
Remember, we're looking for a US airline operating an all Y class configured 727-200. Hmm... whichever airline could it be...?
I was very tempted to reply PSA to this one