Fetching details from the dictionary...
dealer's privilege (of being declared a winner in case of a tie)
「親権」 has another reading with different meanings.
親権を活かして競り勝った。
I leveraged the dealer's privilege and edged out the win.
流局で親権が生きて、もう一局親のまま続いた。
After the exhaust draw, the dealer's privilege stayed alive and the dealer continued another round.
最終局は親権のおかげで勝ちを拾い、優勝に王手をかけた。
In the final round the dealer's privilege let me scrape a win and put us on the brink of the championship.