English is a middle language when it comes to precision of expression—more accommodating of precision than German (for example) but less so than French.
You are right that "get back to work" is a dangling element in your sentence. Usual English convention is to fix dangling elements in one way or another, but exceptions abound. For example,
The doctor you met yesterday observed the patient, instructed the intern, and took notes as he was making his rounds today.
This hardly brilliant English, of course, but the context and construction make it clear enough that "he" and "his" refer to "[t]he doctor you met yesterday" that one probably would refrain from flagging the pronoun as an error.
In your sentence, the parallel use of the word "get" does admittedly suggest that he, not you, is to "get back to work."
On the other hand, good English style accommodates grammatical ambiguity only occasionally, and only when there is no easy way to fix it. There is indeed an easy way to fix it in your sentence, a way you may already have noticed:
You should tell him to get up and to get back to work.
Though your general question is valid, your specific sentence should be fixed in this way.