The former Chelsea midfielder was appointed as the Cottagers' first-team coach in 2019 and during his two years in charge saw them promoted to the Premier League and then relegated the following season.
It is proving to be a shrewd appointment by the Cherries who currently sit top of the Championship after nine games played.
Speaking exclusively to talkSPORT, Parker said: "Firstly, I didn't just leave Fulham, what happened in the summer was mutual.
"I know the word mutual is often put there and everyone works out that, from normally one side of the party, it isn't [mutual] well that was the case at Fulham.
"It was a conversation at the end of last year, after a disappointing year, where obviously, being there for two years, I felt there were certain things needed to happen.
"Certain things that did happen obviously made my decision in terms wanting this. And what I felt was best was for the club to move forward.
"On the flip side I also understood that, as a football club, the investment that one football club has put in to certain players or contracts and stuff like that, that might not be as easy as that.
"I understood both situations and I felt that it was best for myself for me to able to get the best out of Fulham, for me to be successful there and the team and Fulham to be successful, unless it was under certain things that I wanted I didn't think it was going to be the best for us.
"Fulham felt different and of course that's their prerogative and I understood why they see it like that so we both decided that was the best way to go.
"I was in a fortunate position that Bournemouth was a job that was here and ready.
"Bournemouth showed real keen interest in me to come here, sold me how they see it. Did I see it as a very good fit for me? Yes, definitely, and hence why I am here really.
"I don't think it was as straight forward as me just walking out on Fulham and choosing Bournemouth.
"Do I feel that I've come to a football club that suits and probably, in this moment in time, is perfect for me and my squad and around an ownership team that feel we can progress? Definitely.
"That's why it happened really."