Targets that are immune to paralysis may still be staggered.
Although Paralyze is an Expert-level spell and only supposed to be in random loot at around levels 35+, rarely it can be found in "boss chest" loot at levels 30 and under.
Dragons, Dwarven automatons, Ghosts, and most hovering creatures are entirely immune to Paralyze and all other forms of paralysis.
Fortify Alteration potions increase the duration of the Paralyze spell. If you utilize the Fortify Restoration glitch, it is possible to make Fortify Alteration potions which will vastly increase the duration of the Paralyze spell, and any other Alteration spell, even without the Stability or dual casting perks.
If you Soul Trap a paralyzed target, they may get up and play their walking animation upon death. ?
Killing someone who is paralyzed may result in them not being able to be looted. To avoid this bug, you can paralyze someone and then attack them until their health is very low, then when they begin to stand back up, land the killing blow/spell. ?