The 27-year-old, who was born in Oslo, signed a three-and-a-half year deal.
"Jonathan is a class player who fits perfectly into our club and style of play," manager Jostein Flo told the club website.
"He is good with both feet, very capable and is a player with a lot of energy who will delight the fans here at Marienlyst."
Parr's contract with Championship side Ipswich was due to expire in the summer.
Ipswich midfielder Jonathan Parr has completed his move to Norwegian side Stromsgodset.