Parker said today he and his family had been placed in a "terrible position".
"This has been going on for ages now, and it takes a toll," said Parker in a statement, provided by his lawyer Michael Heron QC.
"I haven't been charged with anything and it doesn't feel right that my name will be connected to matters I've had no chance to defend myself against."
He said he could not imagine anyone in his position not wanting to seek name suppression.
"It's a terrible position to be in," he said.
"Things have been said about me that are quite untrue; some of them have already been proven untrue, and I was given no chance to respond at the time they were made.
"It's caused a lot of stress and worry for myself and my family – and it just seems totally wrong.
"Having said that, I need to accept the decision and get on with my life. I have other fights to fight."
Parker is currently in the UK, having flown out last weekend to start preparations for his expected next bout, against Derek Chisora, possibly in early May. Parker beat fellow Kiwi Junior Fa at Spark Arena earlier this month.
Heron said Parker and his legal team "respectfully acknowledge and accept" the court decisions that had led to name suppression lapsing, "whilst disagreeing with them".
"In 2019, Mr Parker applied for name suppression on the basis his name would be referenced in a High Court trial in which three men were charged with, and eventually found guilty of, the importation and supply of methamphetamine," said Heron.
"Today, his pursuit for continued name suppression has drawn to a close, the Supreme Court declining him leave to appeal the decisions of the High Court and Court of Appeal, both of which ruled that, notwithstanding errors made, it was in the interests of open justice that name suppression be declined.
"In summary, Mr Parker was investigated by the police, who found there was insufficient evidence on which to charge him. The police made a public statement confirming that, after questions were raised with them."
He said Parker had denied, on oath, any involvement with methamphetamine or the importation of drugs.
"He has not engaged in that conduct. Despite the lack of evidence against him, unfounded allegations were made against Mr Parker in court," said Heron.
"Those allegations were not supported by evidence [and] were unnecessary and were unfair. Unfortunately, those matters were beyond the control of Mr Parker."
Heron said that in New Zealand, name suppression was essentially the only protection available for people placed in this position.
"However, openness is a fundamental aspect of our system and it is a question of judgement as to where the balance should be struck between competing interests."
Heron said he had advised Parker not to comment further. He had accepted that advice, he said, and "resolved to put this matter behind him and concentrate on the future".
