stablemaster who owns his own stable
In sumo, a stablemaster who personally owns the training stable (heya), as opposed to one who manages a stable owned by the sumo association.
A stablemaster who owns his own stable is responsible for managing his own heya.
親方 is a general term for a master or boss, while 部屋持ちの親方 specifically refers to a sumo stablemaster who owns the stable.
Compound of 部屋持ち (heya-mochi, 'owning a room/stable') + の (possessive particle) + 親方 (oyakata, 'master'). The term is specific to sumo culture.