Parallel Quests can be triggered by talking to the robots in the building of the same name. There are Online and Offline Parallel Quests. You need a Playstation Plus subscription in order to take Online PQs. Offline PQs are readily available and more will be unlocked as you complete preceding PQs and Time Patrol quests.
PQs are very important since they're the main source of experience and zeni from the game. Since you can repeat any PQ as many times as you want, it's possible to get multiple copies of items, and skills. For starters, most PQs are accepted from this building in the Time Machine Sector.
Additional PQs are unlocked as you complete preceding PQs or complete Sagas (story missions). There are also PQs that can only be triggered by talking to various characters (like Raditz or Nappa for example) hanging around near the Time Machine Sector. After completing their PQs, these will be conveniently listed in the Quest Information building for you to replay them again easily.
There are hidden conditions for every PQ. If you haven't discovered the hidden condition yet, it will be marked as blue "??????" in the quest details page. Unlocking these hidden conditions will trigger special events, additional enemies, and extra locations which will allow you to get a higher rank at the end of the mission. The individual pages on this guide will feature all the hidden conditions; however following the conditions doesn't seem to guarantee to trigger them all the time.
PQs have a list of possible rewards. For completing each PQ, you're guaranteed to earn experience and zeni. Costumes, and skills however are only dropped by chance and by defeating opponents. If you're after a specific skill, depending on your luck, you may get it on your first run or after several attempts. Also, you have to finish the quest until the end in order for you to keep the collected items in the previous scenarios / objectives.
Before heading to a PQ, always remember to do the following: