The 23-year-old featured in the Robins' friendly defeat to Rangers and travelled to Spain for the club's training camp along with fellow trialist Kelle Roos.
But head coach Sean O'Driscoll made a decision to plump for Parish, who put in a memorable display in Saturday's 1-0 win over Forest Green, as his long-term choice.
"Elliot is 23, he has already been a back-up goalkeeper in the Championship and knows what it's all about, so he's a better option for us," he told Bristol City Player.
"Our priority was to get a number two in and Elliot fitted the bill."
A club statement confirmed that O'Driscoll is hopeful of concluding a deal in the near future.