Both the Paralyzed and Stunned conditions include the following:
- a [paralyzed/stunned] creature is incapacitated (see condition) and can't move (or in the case of paralyzed speak)
- The creature automatically fails Strength and Dexterity saving throws
The Incapacitated condition specifies:
- An incapacitated creature can't take actions or reactions
Some creatures, such as Dragons, have Legendary Resistance:
Legendary Resistance (3/Day). If the dragon fails a saving throw, it can choose to succeed instead.
Legendary Resistance does not require an action or reaction, so the Incapacitated condition does not seem to prevent its use.
RAW, there doesn't seem to be anything preventing an incapacitated, paralyzed or stunned creature from "choosing" something that doesn't explicitly require any kind of action.
[Perhaps the question is moot anyhow because if a creature had legendary resistances left, they probably would not allow themselves to be incapacitated.]