Rod Staffwand
aka Ermlaspur Flormbator
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.
No error.
Ability Checks, PH p174.
"An ability check tests a character’s or monster’s innate talent and training in an effort to overcome a challenge. The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results."
Contests, PH p174.
"Both participants in a contest make ability checks appropriate to their efforts."
When you're paralyzed, petrified or unconscious you can't take actions or reactions, move or speak. That makes making or resisting Strength and Dexterity checks pretty much impossible.
Both RAW and RAI are clear in this instance.
Interestingly, just being incapacitated means that you can move and speak but not take actions or reactions. I can't remember if any spells or effects cause the incapacitated condition on its own but it would seem to be a groggy, dazed state in which you might mumble a few words and draw your sword. I'd let such an individual contest a grapple, albeit at disadvantage.