The 30-year-old, who was a free agent after leaving West Brom, has signed a two-year deal with Ian Holloway's side
On Saturday, Palace manager Ian Holloway claimed that Thomas had agreed terms, but the club have yet to officially announce the capture of the 30-year-old.
Parish, though, says that Thomas had definitely signed and was already at the club's pre-season training camp in Portugal.
Parish told the Croydon Advertiser: "The paper work has gone through, he's training with us in Portugal so it's all done.
"I'm not sure why it hasn't been announced yet though. It's a two-year deal for Jerome."
Parish revealed that Thomas, who was a free agent after his West Brom deal expired, was keen to link up with Holloway again, having played under him in a loan spell at QPR in 2001.
And Parish believes the presence of Holloway played a big role in his decision to sign for Palace.
"We’ve been talking to him for a while, he's absolutely fine and he loves Ian Holloway," Parish said.
"He's worked with Ian before – he's a big draw for Palace. If Ian has worked with a player before, typically they've got a very good and positive view about coming back to play for him.
"It's definitely a plus for us I think. Then other players ring other players who have worked for Ian before and it helps.
"I'm sure he had offers elsewhere – he's a Premier League football player."