Yes, on Joint Business routes at least, BA charges the YQ/YR that applies to paid fares on partners. AA also charges BA's YQ to AAdvantage members if they book BA.
BA/AA charge the exact same YQ/YR amounts on trans-Atlantic flights. BA collects it on all bookings, paid or rewards, AA collects it only on paid bookings and BA awards. AA doesn't charge it to AA members on AA rewards. Has been thus since the Joint Business started in 2008, but of course the surcharges keep going up…