Should a paraphrase go in quotation marks?
“No.”
Feel feel to paraphrase me. 
John said the Nissan Sentra was economical and reliable. = paraphrasing
John said, “The Nissan Sentra is economical, reliable and shown to be one of the more dependable cars on the road.” Direct quote" " "quotation needed.
If you’re paraphrasing **within **a quotation, you need to use brackets, as in:
He said, “We visited [three family members] before returning home.”
Note that if you are transcribing an interview you are allowed to clean up the er/ahs and grammar a wee bit as long as you don’t change the essential meaning.
I often see quotations (inside quote marks) that were obviously made in another language and therefore have been translated. Somehow that is considered legitimate, but isn’t it really a form of paraphrase?
I think panache45 gave a good example.
He/she said: “He said, “We visited [three family members(of panache45’s orginal reference)] before [going back] home.””
In a sense, yes, but the idea is that a direct quotation is to render into the language-of-translation as accurately as possible what was said in the language-of-origin. There are times when this is impossible: werden and “become” are not precise equivalents; to render the contrast of savoir and connaitre into English requires either paraphrase or a synonym that weakens the impact. Large aspects of Russian idiom and syntax do not translate directly. But a translation rendered as direct quotation should be as close as the translator can reasonable come to what was said in the original.
Now, if we could get FOX News to understand that difference!!