My Latin teacher was talking about parataxis and hypotaxis using coordinating vs subordination conjunctions. He said that may have been the way people spoke in the past. I don't believe there ever was or is a language that is purely paratactic.
Is there any example of languages past or present that do not use subordinating clauses
First, your question at the end doesn't really make sense, because conjunction is not subordination. I think you're asking for languages without subordination, so you shouldn't add "or conjunctions".
So to what I think is your actual question: Are there languages without subordination? Pirahã is claimed to be such a language.
And of course your Latin teacher is right when saying that "that may have been the way people spoke in the past", because "that" here refers back to "parataxis and hypotaxis using coordinating vs subordination conjunctions".
Apparently, all languages signed or spoken have some form of syntactic subordination. However, languages differ significantly in the types of subordinata constructions present, in how heavily they use subordination and for what functions. For instance, it has been noted that subordination is often weakly developed in polysynthetic languages.
So in some sense ``this is how people speak'' may well be the case---which does not excluse existence of subordination in a language.
