Parish, formerly of Aston Villa and Cardiff, travelled with the Robins on their pre-season trip to Spain.
The 23-year-old will now provide competition to fellow summer signing Frank Fielding for the number one jersey at Ashton Gate, and he is also likely to be a regular starter for Mark O'Connor's Under-21 side.
"He fits the financial model we're working to and we're pleased to get it sorted," City boss Sean O'Driscoll told the club's official website.
"He's a good size, a good keeper and one where there's room for improvement. He wants to push Frank and I'd have no hesitation putting him in the team."
Parish added: "I'm delighted to sign for the club, having been here for a number of weeks now.
"It's a great club and although there was disappointment with relegation last season, everything is geared up to bounce back strongly.
"My aim is to challenge Frank hard for the number one jersey; he's a great keeper and I'll learn a lot working alongside him."