The New Zealand boxer is preparing for a crucial fight with Derek Chisora next month and has tapped into Fury's knowledge in a bid to rejuvenate his career.
Parker told Sky Sports: "I called Tyson and said: 'I'm looking for a trainer.'
"He said Andy would be perfect. Five days later I was on a plane flying to Ireland. Everything happened fast."
Parker has spent time training with Fury and Lee and believes the approach is exactly what he needs after losing his world heavyweight crown.
He said: "It's exactly what I need for Chisora. It's probably what I needed for Dillian Whyte, to not be bullied and chased and thrown around the ring.
"I didn't prepare that way for the Whyte fight. But I should for this fight."
Parker recently parted ways with Kevin Barry after a long-term partnership and felt that it was time to take a different approach.
He said: "I was with Kevin for eight years so became used to his way of doing things but Andy has shown me a few other things. We clicked from the beginning.
"I was so used to the training we had, flying to Vegas, the same routine. I'm not getting any younger. I want to achieve in boxing. Kevin and I decided it was time to try something else.
"This is a new excitement, it is new and different. I'm challenging myself."