The 26-year-old joins fellow summer arrivals Cheikhou Kouyate, Max Meyer and Vicente Guaita at Selhurst Park as manager Roy Hodgson bids to avoid the relegation fears that dogged the club for most of last season.
"We are very pleased," Parish told Sky Sports News. "The manager was keen to do it obviously, and we have got more goals in the team.
"He got eight Premier League goals [for Swansea], and that really is what we were looking for.
"He has got quality on the ball as well, so we are very excited to do that, and all told as we sit here today, we are pleased with the business we have done."
The Eagles face newly-promoted Fulham when their Premier League campaign gets underway on Saturday.