After completing the Return of The Saiyans story line you will have access to the 7 that you need for this achievement.
These Parallel Quest are;
Parallel Quest Tutorial
World Tournament Tag Team
Prepare For The Attack Of Saiyans!
Saiyan Blood - This quest is unlocked by talking to Raditz in the tower, he will be in the Time Machine Station area.
Saibamen's Revenge
Attack of The Saiyans
Invade Earth - This quest is unlocked by speaking to nappa in the tower, he will be in the time machine station area.