To unlock the Bardock and Broly Saga, find all five shards as indicated as follows. You must have completed the story line first.

Complete Quest 3 in the Demigra Saga.
Speak with Recon at Time Square.
Complete Parallel Quest 49.
Defeat Goku in Parallel Quest 14, or defeat Ginyu in Goku's body in Parallel Quest 18.
Complete Parallel Quest 2, or defeat Super Saiyan Goku in Parallel Quest 21.

After obtaining all the shards, talk to Trunks to begin the secret Saga mission. Complete the saga and "Close Calls Are More Exciting!" will be earned.