Forsberg will become an unrestricted free agent July 1. The 33-year-old Swedish superstar was playing out the final months of a two-year US$11.5-million contract.
The former MVP, who's been slowed this season by a chronically troubled right foot, had 11 goals and 29 assists for 40 points in 40 games for the Flyers.
Upshall, 23, a left-winger from Fort McMurray, Alta., had two goals and an assist in 14 games for the Predators. Parent, 19, a six-foot-two defenceman from Prince Albert, Sask., was the Predators' first round pick from the Ontario Hockey League's Guelph Storm in the 2005 NHL Entry Draft.
Forsberg had a meeting with Flyers chairman Ed Snider on Sunday to discuss his future, but declined to comment the next day about the specifics of the meeting and refused to say whether he asked to be traded.
He had said that he didn't want to address his future until he solved the issue with his right foot.
The foot injury kept him out 16 games this season, but since returning from an all-star break Forsberg has three goals and nine assists in nine games.
On Monday night, Forsberg scored the go-ahead goal in the third period of the Flyers' 6-1 win over the Detroit Red Wings.
The NHL trading deadline is Feb. 27.