These are special, challenging missions that you can tackle once you’ve reached a certain part of the story (month) and with the required units recruited in your army. Completing these missions are really recommended since they’ll give you a lot of valuable rewards like Hero relics, unique weapons, Battalions, Renown, and many more.
Paralogues require one Battle Activity Point and will expire after some time. Clear them as soon as they’re available to take advantage of the rewards they offer. They also give you some light on the involved characters’ background stories you wouldn’t have been privy to otherwise.