Parish has been brought in to ease Gary Waddock's keeper crisis after first-choice stopper Nikki Bull was ruled out of action with a thigh strain.
Matt Ingram is also unavailable after undergoing surgery on Monday to repair damaged cartilage in his knee.
Parish, 22, started his career at Aston Villa and had loan spells with Lincoln and Cardiff last season before joining the Bluebirds on a permanent basis in January.
"Bully's been struggling for a while now but he's an important player for us and we wanted him out there," Waddock told the club's official website.
"However, it's come to the time now where he needs rest so it can heal properly, but we're hopeful it's only a short-term problem."