FormerlyHemlock
Hero
You don't auto-fail Strength and Dexterity checks while paralyzed--you can't make them at all. If you're paralyzed you're incapacitated as well. If you're incapacitated you can't take actions. If you can't take actions you can't make checks. It's as simple as that. You can't contest a grapple while paralyzed or petrified since you can't make a check to do so.
The bold line is the error. Contests can be triggered by the actions of others. For example, under DMG Disarm rules, you can attempt to disarm another creature of its weapon. You make an attack roll, opposed by the other creature's Str (Athletics) or (IIRC) Dex (Acrobatics). The other creature does not have to spend an action or reaction to resist, it is built in to the Disarm attempt. I agree that the other creature should auto-fail this check if paralyzed, hence the ruling in the OP.