Eden Park is be considered to host the potential bout against Mexican American Andy Ruiz Junior.
Fury had been due to fight Wladimir Klitschko in Manchester on October 29, defending the WBA and WBO heavyweight belts he took off the Ukrainian last November.
One attempt at a rematch had already been abandoned because of an ankle injury sustained by Fury, and his management cited medical reasons behind the latest cancellation last week.
A source close to the planned fight told PA Sport he had been informed "traces" of cocaine were found in Fury's system. The 28-year-old fighter's camp would not confirm that.
ESPN reported Fury was tested by the Voluntary Anti-Doping Association (VADA), which is based in Las Vegas, as part of the routine agreed by both fighters before the fight.
Tonight's Parker fight, which could propel the 24-year-old into a potential title challenge against Anthony Joshua or Andy Ruiz Jr, or force him to re-build his career, has been distributed to 53 countries around the world, and will represent the largest boxing broadcast out of New Zealand.
The heightened interest has been sparked by the fact the undefeated Parker is on the rise and that his opponent, a giant 2.01m Ukraine, is European. With Parker the mandatory challenger to Brit Joshua's IBF title, and potentially also closing on a home fight against Mexican-American Ruiz Jr for Tyson Fury's WBO title, the South Aucklander is becoming a lucrative prospect.