Nope! Nintendo has confirmed that only one player needs the Booster Course Pass to play online in Friends and Rivals races:
Courses from wave 1 of the Mario Kart 8 Deluxe – Booster Course Pass can be played from March 18th locally or online in Friends and Rivals races, even if only one player owns the Booster Course Pass or has access to it as part of a Nintendo Switch Online + Expansion Pack membership. From March 22nd, courses from wave 1 will also appear in the Random selection when playing Global or Regional races with anyone online.