The paralyzed condition states:
A paralyzed creature is incapacitated (see the condition) and can’t move or speak.
And for incapacitation:
An incapacitated creature can’t take actions or reactions.
Compare this to the 3.5 version, which specifically mentions physical actions:
This special attack renders the victim immobile. Paralyzed creatures cannot move, speak, or take any physical actions. The creature is rooted to the spot, frozen and helpless. Not even friends can move his limbs. He may take purely mental actions, such as casting a spell with no components.
Does this mean that, counterintuitively, by rules as written in 5th Edition a paralyzed creature cannot take actions that don’t require movement (e.g. casting a spell with no components)? It seems a silly interpretation, but that’s what the rules seem to be saying (assuming this resource accurately reproduces the rules).