According to a page on the Northern Illinois University site, the festival concludes a Novena (nine days) of prayer.
The festivity commemorates the search of the Holy Cross by Queen Helena (Reina Elena) and her son, the newly converted emperor Constantine. After the Holy Cross was found in Jerusalem and brought back to rome, there was a joyful celebration for thanksgiving.
Nine days of prayer (a novena) in honor of the Holy Cross precedes the Flores de Mayo or Santacruzan. This festival was introduced by the Spaniards in the Philippines and has since become part of Filipino traditions identified with youth, love and romance.
The feast of the Finding of the True Cross as introduced by the Spanish would have been celebrated on May 3, which is when it appeared in the Galician calendar. The festival was suppressed when it was combined with the Roman celebration of the Triumph of the Cross (September 14) in Pope St John XXIII's New Calendar of 1960.
As the main procession in New York always occurs on a Sunday, the Novena of prayer would start nine or ten days prior to that — depending on whether the procession is on the last day of the novena or the day following it, celebrating its completion. This year, it fell on May 18; thus the novena would have started on either Friday 9 May or Saturday 10 May.
I would surmise that the novena starts on the Friday or Saturday following May 3, which would fix the final festival on the third Sunday in May.