Or you could look at melee parry as parrying a brawling or weapon attack, regardless if you have a weapon, as you described the mixed-martial arts.
Melee and brawling are similar, but someone trained in melee parry (bare hand and/or weaponed attacks) would still be good against brawling attacks, but not necessarily the opposite.
Maybe if attacked by a weapon, you must use melee parry, but if it is bare handed, you can choose between melee/brawling (unless you are using a weapon, then melee is required)