The play invigorated Ginobili, who had two steals and a 3 to bridge the third and fourth quarters and tie the game at 84.
Hunter refused to blame the foul for Phoenix's struggles in the fourth.
"We talk about being a no-excuse organization and that's no excuse," he said. "We're not at that level right now, but I can tell you this, it won't be long before we are."
Just as it did in Wednesday's 106-102 victory over New Orleans, San Antonio struggled early.
The Spurs missed their first three shots while failing to score in the first 3 minutes. After Danny Green and Kawhi Leonard each missed a 3, Parker scored on a layup after Goran Dragic fell while attempting to backpedal defensively.
The basket sparked the Spurs, who made their next 10 attempts in a 22-13 run. Parker scored 10 points during the run, including a layup that put San Antonio up 22-19 with 4 minutes left.
Beasley scored 10 points in the second quarter, including six during an 11-6 surge that put Phoenix ahead 49-48 with 4:21 left in the half.
"They have history," Beasley said. "They just know how to win. The same guys have been here for the last hundred years. They know how to play with each other. They've won titles on top of titles. We've got to be able to take care of the ball, at the very least get a shot."
NOTES: Duncan has sat out four games this season after missing eight during last year's strike-shortened season. ... Suns F Channing Frye (enlarged heart) remains out indefinitely. Frye has not played this season. ... The Spurs swept the season series against the Suns last season, winning the four games by an average of 8.5 points. ... Phoenix general manager Lance Blanks spent five seasons in San Antonio's front office, ascending to director of scouting before joining the Suns.