I believe this was debunked as I thought the same thing - Rask signed the offseason before that cup run
Not the greatest source, but Wkipedia states:
"On 28 June 2012, Rask re-signed with the Bruins to a one-year, $3.5 million contract. Prior to the declaration of the
2012–13 lockout, Rask was named as the starting goaltender for the Bruins, replacing Tim Thomas, who would eventually be traded to the
New York Islanders on 7 February 2013.
[24] During the lockout, which ended on 6 January 2013, Rask played for
HC Plzeň, which won the
Czech Extraliga that year. After the NHL resumed play, Rask led the Bruins to their second Stanley Cup finals in three seasons in the 2013 playoffs. In the third round of the playoffs against the top seeded
Pittsburgh Penguins, Rask faced 136 shots in four games played, allowing two goals while making 134 saves for a 0.50 GAA and a .985 save percentage. In the
Stanley Cup Finals, the Bruins were defeated in six games by the Presidents’ Trophy-winning Chicago Blackhawks, as Rask registered a .932 save percentage.
[25] On 10 July 2013, the Bruins re-signed Rask to an eight-year, $56 million contract."