Ismail Ali scored 19 points, Rasheed Worrell added 13, and Antwon Lillard and Zach Denny scored 12 apiece for the Falcons (11-5, 2-1 Mid-American Conference), who made 22 of 30 free throws.
Trailing by five at halftime, Bowling Green rallied to lead 51-46 after a 14-3 run in which Parker scored eight points.
Ohio closed to within 60-56 with 10:48 left on a 3-pointer by Jordan Dartis, but Bowling Green pulled away on a 9-4 spurt in which Parker hit a layup and 3-pointer.
Ali made five free throws in the final 2:49 and the Falcons outscored the Bobcats 9-6 down the stretch.
Bowling Green won the rebound battle 42-32 and the Bobcats were hurt by 30 personal fouls.
Kenny Kaminski scored 15 points, Antonio Campbell and Jaaron Simmons added 13 each, and Dartis had 11 for the Bobcats (10-5, 1-2).