The paralyzed condition notes
A paralyzed creature is incapacitated (see the condition) and can't move or speak.
But is a monster 'speaking' when it's using telepathy? Can it use it while paralyzed?
The rules for monster telepathy specifically cover this case:
A telepathic monster can initiate or terminate a telepathic conversation without using an action, but while the monster is incapacitated, it can't initiate telepathic contact, and any current contact is terminated.
A monster that is paralyzed is also incapacitated, so it can't use its telepathy. (If it were stunned, it would also be incapacitated and lose its telepathy, but it would still be able to use mundane speech falteringly.)
However, there's nothing in the telepathy rules preventing an incapacitated creature from replying to another, non-incapacitated, telepathic monster who initiates telepathic contact with it.