Hypotaxis is plain, short English, as encouraged by George Orwell. Mark Forsyth, the author of the book on rhetoric is against it. https://richardswsmith.wordpress.com/2018/01/14/recantation/
Polysyndeton is using lots of conjunctions to keep a sentence going.
“And Jesus took bread, and blessed it, and brake it, and gave it to his disciples saying ‘Take, eat, this is my body.’”
Asyndeton is using no conjunctions.
Parataxis is long, convoluted, complicated sentences full of clauses and subclauses. I associate it most with Proust and Henry James, but the first to write in such a way in English was Sir Thomas Browne, a physician:
“… thus is man that great and true Amphibium, whose nature is disposed to live not onely like other creatures in divers elements, but in divided and distinguished worlds; for though there bee not one to sense, there are two to reason; the one visible, the other invisible, whereof Moses seemes to have left description, and of the other so obscurely, that some parts thereof are yet in controversie; and truely for the first chapters of Genesis, I must confesse a great deale of obscurity, though Divines have to the power of humane reason endeavoured to make all goe in a literall meaning, yet those allegoricall interpretations are also probable, and perhaps the mysticall method of Moses bred up in the Hieroglyphicall Schooles of the Egyptians.”
Dickens used it:
“It was a maxim with Mr Brass that the habit of paying compliments kept a man’s tongue oiled without any expense; and that, as that useful member ought never to grow rusty or creak in turning on its hinges in the case of a practitioner of the law, in whom it should be always glib and easy, he lost few opportunities of improving himself by the utterance of handsome speeches and eulogistic expressions.”
And John Cleland used it in Fanny Hill:
“Coming then into my chamber, and seeing me lie alone, with my face turned from the light towards the inside of the bed, he, without more ado, just slipped off his breeches, for the greater ease and enjoyment of the naked touch; and softly turning up my petticoats and shift behind, opened the prospect of the back avenue to the genial seat of pleasure; where, as I lay at my side length, inclining rather face downward, I appeared full fair, and liable to be entered. Laying himself gently down by me, he invested me behind, and giving me to feel the warmth of his body, as he applied his thighs and belly close to me, and the endeavours of that machine, whose touch has something so exquisitely singular in it, to make its way good into me.”
All of this is taken from Mark Forsyth’s useful and very readable book “The Elements of Eloquence: How to Turn the Perfect English Phrase”