Dart, who had beaten Belgium's Maryna Zanevska in the first round, was broken in the sixth game of the opening set.
Fifth seed Parrizas-Diaz then saved five break points to fend off a fightback from Dart, who let a 40-0 lead in her next service game slip as the Spaniard eventually closed out the set.
Play was suspended at the start of the second set because of rain, but it did not prove a long delay.
Both women failed to hold onto their opening serve games, before the second set levelled up at 2-2 and the rain then returned to force the players off court again, this time for an extended spell.
As the washout continued, tournament organisers announced there would be no play before 6pm local time.
With conditions improving, the match was able to resume but nine hours after commencing the Spaniard finished 6-2 6-4 over Dart.