Dorothy Parker would have to be my favorite poet comedian. Her style of writing frequently includes a wry irony. Witty but not quite comical, her writing often had a caustic element. One of my favorites of hers is a peom entitled "One Perfect Rose."
This is an enjoyable question. If I had to choose one comedian poet, I would choose Plautus, the Roman poet. He wrote in the second century BC and he wrote many poems, which were remakes of old Greek works. The reason why I think he is funny is threefold.
First, he loves inversions. He looks at how society functions and he reverses it. So, slaves overcome masters, women overcome men, children overcome parents, and humans overcome the gods. This reversal contains many funny social situations.
Second, Plautus is a very keen observer of people and society. For this reason, his poetry is very insightful. He can see where the social anxieties are and bring a little humor.
Third, Plautus can be used as a window into ancient society. So, he is a goldmine when he comes to historical information.