Mr Wood, who is a member of the National Party, had discussions with Auckland Future. He supported the intent of the policy but decided not to sign the pledge and stood as an independent.
Another centre-right councillor, Dick Quax, has also declined to sign the pledge.
"I don't want to be restricted from giving citizens of North Shore the focused attention they deserve and need. Also, I am concerned that we must first review the range and standards of service council is providing for our citizens," Mr Wood said.
On Monday, Auckland Future announced it had selected former Takapuna community board member Fay Freeman and Kaipatiki Local Board member Danielle Grant to contest the two North Shore ward seats.
In an electronic newsletter, Ms Barry said she was very enthusiastic about Auckland Future and its pledges.
She said it was encouraging to see someone of the energy and experience of Fay Freeman and Danielle Grant "and I will certainly be supporting them". She also supports Ms Crone.
"To be frank with you I don't really know what George stands for," Ms Barry told the Herald.
"I do know that Danielle Grant and Fay Freeman have four fundamentals that they have as their bottom lines in that pledge and that is why I am supporting them.
"There is an opportunity with Auckland Future for voters to decide with clarity and certainty about the integrity of the people they are voting for," the MP said.
Ms Barry defended her right to support candidates at October's local body elections, saying as a local MP she was concerned with local issues and it was important to be transparent.
"I will work with whoever is elected but I would like to be able to work alongside people who share common values and that is why I feel absolutely comfortable that I am supporting the two local candidates for council and the mayoral candidate," she said.
Mr Wood said he had had calls from National Party members expressing surprise and regret that Ms Barry had breached the tradition of MPs not getting involved in local politics.
Not all National MPs support Auckland Future, running a separate campaign to the other centre-right ticket, Communities and Residents, which ruled the former Auckland City Council for decades.
Auckland Future, said one National MP, was "burning more bridges than they are building".
Tories at war
• Centre-right fragmented with three mayoral contenders and two political tickets
• North Shore MP Maggie Barry supports two new candidates for North Shore ward
• Sitting centre-right councillor George Wood accuses Barry of meddling in local politics